Exploring Parameter Spaces in Coping
with Computational Intractability
Vorgelegt von
Diplom-Informatiker
Sepp Hartung
geboren in Jena, Thüringen
Von der Fakultät IV – Elektrotechnik und Informatik
der Technischen Universität Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
– Dr. rer. nat. –
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender:
Gutachter:
Gutachter:
Gutachter:
Prof.
Prof.
Prof.
Prof.
Dr.
Dr.
Dr.
Dr.
Stephan Kreutzer
Rolf Niedermeier
Pinar Heggernes
Dieter Kratsch
Tag der wissenschaftlichen Aussprache: 06. Dezember 2013
Berlin 2014
D83
Zusammenfassung
In dieser Arbeit werden verschiedene Ansätze zur Identifikation von effizient
lösbaren Spezialfällen für schwere Berechnungsprobleme untersucht. Dabei
liegt der Schwerpunkt der Betrachtungen auf der Ermittlung des Einflusses
sogenannter Problemparameter auf die Berechnungsschwere. Für Betrachtungen dieser Art bietet die parametrisierte Komplexitätstheorie unsere
theoretische Basis.
In der parametrisierten Komplexitätstheorie wird versucht sogenannte Problemparameter in schweren Berechnungsproblemen (meistens, gehören diese
zu den NP-schweren Problemen) zu identifizieren und diese zur Entwicklung
von parametrisierten Algorithmen zu nutzen. Parametrisierte Algorithmen
sind dadurch gekennzeichnet, dass der exponentielle Laufzeitanteil durch eine
Funktion abhängig allein vom Problemparameter beschränkt werden kann.
Die Entwicklung von parametrisierten Algorithmen ist stark motiviert durch
sogenannte heuristische Algorithmen. Entgegen der theoretisch begründeten
Berechnungsschwere weisen heuristische Algorithmen auf praktisch relevanten
Probleminstanzen ein oft effizientes Laufzeitverhalten mit einer akzeptablen
Lösungsqualität auf. Die dafür plausibelste Erklärung ist, dass praktische
Instanzen nur selten den jeweils „ungünstigsten“ Instanzen entsprechen, sondern oft gewisse anwendungsspezifische Charakteristika aufweisen. Eines der
bekanntesten Beispiele dafür ist das sogenannte „Kleine-Welt-Phänomen“,
welches bei der Analyse von sozialen Netzwerken beobachtet werden kann.
Dabei gilt kurz gesagt, dass obwohl jeder Knoten in einem solchen Netzwerk
nur wenige direkte Verbindungen hat, die Länge eines kürzesten Pfades
zwischen je zwei Knoten in der Regel höchstens zehn ist. Die Identifikation solcher Problemparameter und das Ausnutzen der Strukturen, welche
durch kleine Parameterwerte impliziert werden, ist das Hauptanliegen der
parametrisierten Algorithmik.
In dieser Arbeit werden drei Ansätze zur Identifikation dieser Problemparameter vorgestellt und sie werden exemplarisch auf vier NP-schwere
Probleme angewendet. Jeder dieser vier Ansätze führt zu einem sogenannten
„Parameterraum“ (engl. parameter space). Dieser bezeichnet eine Menge
von Problemparametern, welche untereinander in Beziehung stehen in einer
Art und Weise, welche das automatische Übertragen von Härteergebnissen
(z. B. die Nichtexistenz eines parametrisierten Algorithmus) aber auch von
Machbarkeitsergebnissen (z. B. Existenz eines parametrisierten Algorithmus)
3
erlaubt. Damit ermöglicht der Begriff des Parameterraumes eine systematische und strukturierte Darstellung der parametrisierten Komplexität und
erleichtert die Identifikation jener Strukturen, welche maßgeblich die Berechnungsschwere des jeweiligen Problems bestimmen. Nachfolgend werden die
behandelten Probleme und die dazugehörigen Ergebnisse hinsichtlich der
parametrisierten Komplexität beschrieben.
Der erste Ansatz zur Identifikation von Problemparametern ist spezialisiert
auf Graphprobleme und studiert diese, im allgemeinen NP-schweren Probleme, in ihren Einschränkungen auf speziellen Graphklassen. Dieser Ansatz
wird durch die Anwendung auf das Metric Dimension-Problem vorgeführt.
Metric Dimension ist das Problem für einen gegebenen Graphen eine
Knotenteilmenge auszuwählen, sodass alle Paare von Knoten sich in ihrer Distanz (Länge eines kürzesten Pfades) zu wenigstens einem der ausgewählten
Knoten unterscheiden. Wir beweisen mittels einer parametrisierten Reduktion von einem W[2]-schweren Problem, dass Metric Dimension keinen
parametrisierten Algorithmus für den Parameter Lösungsgröße besitzen kann
(unter der Annahme dass W[2] 6= FPT). Insbesondere gilt dieses Härteergebnis auch auf der stark eingeschränkten Graphklasse sogenannter bipartiter
Graphen mit Maximalknotengrad drei. Durch die erwähnte Reduktion wird
zusätzlich ein Inapproximationsergebnis impliziert, welches besagt, dass es
keine Polynomialzeitapproximation mit einem Faktor von o(log n) geben
kann, es sein denn P = NP. Diese untere Schranke beweist, dass aus der
Literatur bekannte Approximationsalgorithmen bis auf konstante Faktoren
optimal sind.
Der zweite Ansatz zur Identifikation von Problemparametern untersucht
den Einfluss sogenannter struktureller Parameter auf die Berechnungsschwere
eines Problems. Im Gegensatz zum ersten Ansatz, in welchem das Verhalten
bei Nichtexistenz gewisser Graphstrukturen untersucht wird, ist die Idee
hier, die Häufigkeit des Auftretens gewisser Strukturen als Parameter zu
betrachten. Der Ansatz der strukturellen Parametrisierung wird, insbesondere auch zum Nachweis seiner universellen Anwendbarkeit, anhand zweier
Probleme diskutiert. Im Detail werden ein Graphproblem namens 2-Club
und das Zahlenproblem Vector (Positive) Explanation betrachtet.
Bei 2-Club sucht man in einem gegebenen Graphen einen großen Teilgraphen mit Durchmesser höchstens zwei. 2-Club besitzt Anwendungen
in der Analyse von sozialen und biologischen Netzwerken. Der untersuchte
Raum der strukturellen Parameter für 2-Club ist sehr umfangreich und
beinhaltet neben Graphparametern wie Baumweite und degeneracy auch Parameter welche sich aus dem Paradigma der „Distanz zu einfachen Instanzen“Parametrisierung (engl. distance from triviality parameters) ableiten. Wir
beweisen unter anderen, dass 2-Club NP-schwer ist selbst wenn die Distanz
(Anzahl Knotenlöschungen) zu bipartiten Graphen eins ist oder die dege-
4
neracy fünf ist. Für den Parameter h-index, welcher insbesondere aus der
sozialen Netwerkanalyse motiviert ist, zeigen wir W[1]-Schwere. Neben diesen
Härteergebnissen präsentieren wir auch mehrere parametrisierte Lösungsalgorithmen für 2-Club. Unter anderem wird ein parametrisierter Algorithmus
für den Parameter „Distanz zu einem Cograph“ und für die Baumweite angegeben. Wir beweisen weiterhin (unter der Annahme der Strong Exponential
Time Hypothesis), dass es für den Parameter „Distanz k 0 zu einem 2-club“
0
keinen Algorithmus mit Laufzeit O((2 − )k · nO(1) ) geben kann und deshalb
0
ein bereits bekannter Algorithmus mit Laufzeit O(2k · nO(1) ) asymptotisch
optimal ist. Weiterhin wird eine Implementierung eines parametrisierten
Algorithmus für 2-Club beschrieben, welcher auf praktischen Instanzen
wie sozialen Netzwerken mit bisherigen Lösungsalgorithmen verglichen wird.
Als zweites wird eine Anwendung des strukturellen Parametrisierungsansatzes auf das Vector (Positive) Explanation Problem diskutiert. Das
Vector (Positive) Explanation-Problem ist für einen gegebenen Vektor
von natürlichen Zahlen eine Menge von „homogenen Segmenten“ (spezielle
Vektoren) zu finden, sodass die Segmente sich zum Eingabevektor summieren.
Die Betrachtung von strukturellen Parametern wird insbesondere durch eine
einfache Datenreduktionsregel motiviert, deren Anwendung die Parameter
„Lösungsgröße bzw. Anzahl k der Segmente“ und „Anzahl n der Vektoreinträge des Eingabevektors“ zueinander in Beziehung setzt: Konkret kann man
für alle Instanzen k < n < 2k annehmen. Außerdem wurden strukturelle
Parametrisierungen bereits in früheren Arbeiten zu diesem Problem benutzt,
z. B. bei der Entwicklung von Approximationsalgorithmen. Konkret geben
wir einen parametrisierten Algorithmus für den Parameter „größter Abstand
zwischen zwei aufeinanderfolgenden Zahlen im Eingabevektor“ an. Motiviert
durch die Beziehung k < n < 2k werden Parametrisierungen dem Paradigma
der „Distanz zu einfachen Instanzen“ folgend betrachtet. Wir beweisen, dass
das Vector (Positive) Explanation-Problem sogar im Falle n − k = 1
NP-schwer ist. Für den Parameter 2k − n weisen wir nach, dass die Problemvariante Vector Explanation einen parametrisierten Algorithmus zulässt,
die Vector Positive Explanation-Variante jedoch W[1]-schwer ist.
Der dritte Ansatz betrachtet den Einfluss verschiedener Nachbarschaftsstrukturen auf die parametrisierte Komplexität von Algorithmen, welche auf
dem Prinzip der lokalen Suche basieren. Grundprinzip der lokalen Suche
ist, dass eine bereits gefundene Lösung durch eine Verbesserung, welche
innerhalb einer bestimmten Nachbarschaftsstruktur gesucht wird, ersetzt
wird und dadurch die Lösung schrittweise optimiert wird. Wir untersuchen
den Einfluss verschiedener Nachbarschaftsstrukturen auf eine lokale Suchvariante des bekannten Traveling Salesman-Problems, LocalTSP genannt.
Es wird eine Modifikation eines bereits bekannten W[1]-Schwere-Beweises
angegeben, auf dessen Grundlage die W[1]-Schwere für einen beschriebenen
5
Parameterraum von Nachbarschaftsstrukturen gefolgert werden kann. Konkret beweisen wir, dass die sogenannte Swap-Nachbarschaft die „schwächste“
Nachbarschaft innerhalb des Parameterraumes ist und LocalTSP bezüglich Swap-Nachbarschaft W[1]-schwer ist. Weiterhin folgt für den Parameter
„Größe k der Nachbarschaftsstruktur“, dass LocalTSP bezüglich der meisten betrachteten Nachbarschaftsstrukturen keinen Algorithmus mit Laufzeit
O(no(k/ log k) ) besitzt (unter der Annahme der Exponential Time Hypothesis). Damit wird die Lücke zu den besten bisher bekannten Algorithmen,
welche eine Laufzeit von O(nk ) haben, im Vergleich zu vorangegangenen
Arbeiten deutlich verkleinert. Den Ansatz verschiedene Nachbarschaftsstrukturen zu betrachten kombinieren wir im Folgenden mit dem ersten
Ansatz der Betrachtung spezieller Graphklassen. Im einzelnen zeigen wir,
dass es für LocalTSP auf planaren Graphen bezüglich der Swap- und
Edit-Nachbarschaft, obwohl auf allgemeinen Graphen W[1]-schwer, einen
parametrisierten Algorithmus gibt.
6
Abstract
This thesis is about systematic approaches to identify tractable cases of computationally hard problems. It studies the influence of specific parameters on
the computational complexity of the problems. Hence, parameterized complexity provides a natural and fruitful framework which forms the theoretical
basis of our investigations.
In a nutshell, parameterized algorithmics deals with identifying parameters
in computationally hard problems (mostly, so-called NP-hard problems) and
exploits them in order to design parameterized algorithms whose (presumably)
unavoidable exponential running time part solely depends on a parameter
value. This approach is strongly motivated by the often observed phenomenon
that, when dealing with NP-hard problems in practical applications, so-called
heuristic algorithms which are tuned to characteristics of the input data are
well-performing, both in terms of running time as well as solution quality.
The most plausible explanation for this behavior is that problem instances
emerging from practical applications are not worst-case instances but rather
admit certain, possibly application-specific, structures. These structures
are often far from being obvious or easy to identify, and in many cases
empirical studies of typical data sets help to discover them. Probably one
of the most prominent examples for this is the small world phenomenon
roughly stating that in social networks, although each node has rather few
neighbors, the length of a shortest path between nodes is often at most ten.
Exploiting small parameter values that are implied by these structures for
the development of exact and “efficient” solving algorithms is the central
concern of parameterized algorithmics.
In this thesis we describe three approaches to identify structures which determine the computational complexity of a problem. These three approaches
naturally lead to so-called parameter spaces, that is, several parameters that
are related to each other in some way which may allow to transfer tractability
and intractability results between them. Hence they pave the way for a
systematic and clear analysis of computational complexity and they help to
chart the “border of intractability”. We next describe our three approaches to
identify tractable cases of NP-hard problems and outline our corresponding
case studies.
The first approach is tuned for graph problems and suggests to consider
the computational complexity of a graph problem, which is NP-hard or
7
parameterized intractable in the general case, when restricted to a special
graph class. In our first case study we prove that the Metric Dimension problem is (presumably) parameterized intractable with respect to
the solution size parameter even on the special graph class of bipartite
graphs with maximum degree three. Metric Dimension is the problem
to select in a given undirected graph a small set of vertices such that any
two vertices differ by their distance (length of a shortest path) to at least
one selected vertex. In terms of parameterized algorithmics, we provide
a parameterized reduction from the W[2]-hard Bipartite Dominating
Set problem to Metric Dimension on bipartite graphs with maximum
degree three. In addition to this parameterized intractability result, from
the same hardness reduction we obtain an approximation lower bound which
matches (up to some constants) the performance ratio of the best known
polynomial-time approximation algorithms. More precisely, we show that for
Metric Dimension on bipartite graphs with maximum degree three there
is no polynomial-time o(log n)-factor approximation unless P = NP.
The second approach is denoted as structural parameterizations of the
input. Compared to the first approach of considering special graph classes
and thus investigating the computational complexity on instances where
a certain structure is “forbidden”, structural parameterization rather deals
with measuring the number of occurrences of a certain structure in the input.
Then the exponential running time part of a parameterized algorithm with
respect to this structural parameters directly depends on how often a “critical”
structure exists. We perform the approach of structural parameterizations
to two problems, a graph problem called 2-Club and a number problem
called Vector (Positive) Explanation.
One of the main applications of 2-Club is the analysis of social and
biological networks. 2-Club deals with finding large subgraphs of small
diameter. The corresponding structural parameter space consists of several
well-known graph measurements such as treewidth (measuring how “treelike” a graph is) and degeneracy (measuring the “sparsity” of a graph). In
addition to these somehow classical graph parameters, in the spirit of the
“distance from triviality” parameterization we consider parameters measuring the distance (number of vertex deletions) to special graph classes on
which 2-Club is polynomial-time solvable. Among others, we prove that
2-Club is NP-hard even if the distance to bipartite graphs is one or the
degeneracy is five. We show W[1]-hardness for the h-index parameter which
is well-motivated from applications in social network analysis. Besides these
computational intractability results, we provide a number of parameterized
algorithms, including a direct combinatorial algorithm for the parameter
treewidth and the parameter distance to cographs. Moreover, we prove that
(unless the Strong Exponential Time Hypothesis fails) an already known
8
parameterized algorithm for the parameter distance k 0 to a 2-club, running
0
0
in O(2k · nO(1) ) time, cannot be improved to an O((2 − )k · nO(1) )-time
algorithm for any > 0. We conclude with some positive empirical findings
with an implementation of some 2-Club algorithms.
We also apply the approach of structural parameterizations to a number problem called Vector (Positive) Explanation. In this way we
demonstrate the versatile usability of structural parameterizations. Vector (Positive) Explanation is the problem to find for a given positive
integer vector with n entries a set of at most k ∈ N “homogeneous (positive)
segments” (vectors with a certain property) that sum to (explain) the input
vector. This problem has applications in data warehousing and cancer radiation therapy. Applying the approach of structural parameterization to
this problem is motivated by a simple data reduction rule whose application
allows to assume that k < n < 2k for all instances. Thus, the natural
parameterization by the solution size n adds little use for a fined-grained
analysis trying to learn which structures cause the computational hardness
of the problem. Extending previous work on the parameterized complexity of Vector (Positive) Explanation with respect to the maximum
value parameter, we prove that Vector Explanation is fixed-parameter
tractable by the stronger (smaller) parameter maximum difference between
consecutive vector entries. Motivated by the relation k < n < 2k, we look
at two “distance from triviality” parameters. We prove that both problem
variants are NP-hard for n − k = 1. To our surprise, with respect to the
parameter 2k − n Vector Explanation admits a parameterized algorithm whereas Vector Positive Explanation is (unless W[1] = FPT)
parameterized intractable.
The third approach examines how the computational complexity of local
search is influenced by different neighborhood structures. In difference
to the first two approaches it rather deals with parameters quantifying
structures in the “solution search space” instead of structures in the input.
The algorithm design paradigm of local search roughly follows the framework
of iteratively trying to improve an already known solution by searching for
any improvement within its neighborhood. Motivated by a parameterized
intractability result for the so-called Edge neighborhood distance we conduct
a systematic study of other neighborhood structures of the local search
variant of the famous Traveling Salesman problem, called LocalTSP.
Traveling Salesman is the problem to find for a given set of objects with
pairwisely known distances a shortest round-tour “visiting” all objects.
In addition to the Edge neighborhood distance, we describe other wellknown neighborhood structures and we arrange all of them into a parameter
space admitting a relation allowing to transfer (in-)tractability results among
them. This allows us, after having identified the “weakest” neighborhood
9
structure with respect to this relation, to prove that the original hardness
proof can be extended to this weakest neighborhood structure. This leads
to a general hardness result for LocalTSP that implies parameterized
intractability (W[1]-hardness) for the entire parameter space. Furthermore,
measuring by a parameter k the size of the local neighborhood we prove
that (unless the Exponential Time Hypothesis fails) there cannot be an
O(no(k/ log k) )-time algorithm for LocalTSP for almost the entire parameter
space and thus we make an important step towards closing the “gap” to the
best known O(nk )-time algorithms. Having shown that on general graphs
LocalTSP does not admit any (parameterized) tractable cases with respect
to most of the considered neighborhood structures, recalling the first approach
to look at special graph classes we examine the parameterized complexity of
LocalTSP on planar graphs. Therein we prove that LocalTSP admits
parameterized algorithms on planar graphs for neighborhood structures such
as Swap and Edit distance.
10
Preface
This thesis summarizes large parts of my work as a research assistant at
Friedrich-Schiller-Universität Jena (from Jan. 2010 till Feb. 2011) and at
TU Berlin (since Feb. 2011), each time in the group of Prof. Rolf Niedermeier
(TU Berlin) who has moved from Jena to Berlin in October 2010. Besides
two introductory and one concluding chapter, this thesis consists of four main
chapters presenting results in parameterized algorithmics for four different
NP-hard problems. All these results were obtained in close collaboration
with several coauthors and large parts are published in conference and
journal papers. Before describing this in more detail, I will list further
research to which I have contributed during these years but which is not
part of this thesis: I have worked in the context of parameterized local
search algorithms [HN10, HN13b], clustering algorithms [Guo+10], lineartime computable kernels [Bev+11], computational social choice [Bre+12],
realization of integer sequences by directed acyclic graphs (DAG) [HN12a],
DAG partitioning [Bev+13], and graph anonymization [Bre+13b, Har+13b].
The first main chapter (Chapter 3) contains work about the Metric Dimension problem which I did in close collaboration with André Nichterlein
(TU Berlin). I attended the talk of Erik Jan van Leeuwen (Max-PlanckInstitut für Informatik, Saarland University) at the ESA’12 conference where
he presented his work on Metric Dimension [Día+12] and André and I already started the same day during dinner to work on one of the mentioned
open questions. Although the basic structure of our first ideas actually
worked at the end, it took us about three months to come up with a readable
draft [HN12b]. Therein, we give a relatively easy to describe parameterized reduction proving W[2]-hardness and inapproximability results. Even
though we worked hard to present it in an appealing way, the correctness
proof of the reduction is quite technical and needs about twelve pages of
argumentation. Actually, this was already a shortened version, as by the
time where we were almost done with writing the draft, I realized that changing the problem from which we reduce from simplifies the argumentation
a lot. I presented the final paper at the IEEE Conference on Computational Complexity (CCC’13) [HN13a]. Compared to this conference paper,
I strengthened the graph class in which the graphs constructed by the reduction are contained from maximum degree three graphs to bipartite graphs of
maximum degree three.
11
Chapter 4 considers the 2-Club problem which deals with finding large
subgraphs of small diameter. Working on this problem was proposed by
Christian Komusiewicz (TU Berlin) who already co-supervised a diploma thesis on this subject [Sch09] (which also led to a short journal paper [Sch+12]).
While we initially tried to answer the question on the lower bound for the
dual parameterization, we quickly realized the rich structure and interesting
behavior of 2-Club with respect to several so-called “structural parameterizations” and we thus started to investigate this systematically. This was also
strongly motivated by the talk given by Bart Jansen (Utrecht University)
at Worker 2011 [Jan11]. As André and I are sharing an office, during this
time I diverted him a lot with “2-Club-questions” and at some point he
got attracted and thus became a member of this project as well. Our joint
work about 2-Club led to two conference publications [HKN12, HKN13].
Although most of the work was developed in close collaboration, Christian
mainly contributed to the kernel for feedback edge set (Section 4.4.2) and
the NP-hardness proof with respect to the distance to bipartite graphs (Section 4.2.2). André was mainly responsible for the NP-hardness proofs with
respect to the clique cover and domination number (Section 4.2.1). I mainly
contributed to the hardness proofs for h-index and degeneracy (Section 4.2.3),
the optimality of the dual parameter (Section 4.5), and I conducted the
experimental results (Section 4.6). While visiting Ondřej Suchý (Czech
Technical University in Prague) we together developed the parameterized
algorithm for the parameter distance to cographs (Section 4.3.1) which solved
a problem left open in our previous work [HKN13]. This algorithm together
with a more detailed version of our foregoing paper [HKN13] form the content
of an extended version [Har+13a].
Chapter 5 contains results obtained together with Jiong Guo (Saarland
University), Rolf Niedermeier, and Ondřej Suchý. Motivated by our work
where we introduced an incremental version of the prominent Graph Coloring problem [HN13b] and the observed relations to the local search algorithm
design paradigm, Rolf Niedermeier and I proposed to work on a local search
variant of the famous Traveling Salesman problem. Main parts of the
work were done during a stay of Jiong and Ondřej in Jena and a trip by
myself to Saarbrücken. Ondřej came up with the technical ideas on how
to extend the W[1]-hardness proof of Marx [Mar08] and I developed the
non-existence proof of polynomial-kernels on planar graphs. We have worked
out the details of most of the other results in close cooperation. We are
grateful to Saket Saurabh (University of Bergen) who pointed Ondřej to a
result in the literature that helped to improve our running time lower bounds
from the conference version of our paper [Guo+11] compared to the journal
version [Guo+13].
Rolf Niedermeier approached me with the idea to work on the Vector
12
Explanation problem (Chapter 6) and I subsequently proposed to investigate it during a group-internal workshop held in March 2012. The problem
is, given an integer vector, to find as few “homogeneous” vectors as possible
that sum to the input vector. While our original motivation was to consider
the more general matrix variant [Kar+11] we immediately noticed that even
the (one-dimensional) vector variant, which is presumably easier, was unstudied for most of the parameterizations that we looked at. Since the vector
variant is of practical interest due to an application in cancer treatment
therapy, and since previous work already demonstrated the usefulness of
structural parameters [Bie+11, Bie+13, LSY07], at some point we decided to
concentrate on the vector case and to systematically explore its complexity
landscape with respect to several structural parameters. I was leading the
discussion and contributed to most results. I presented the resulting paper at
the Algorithms and Data Structures Symposium 2013 (WADS’13) [Bre+13a].
Compared to this conference paper, I enriched the presentation of these
results in Chapter 6 by the introduction of the tick vector and a geometric
interpretation of the problem which both help to simplify and illustrate the
technical difficulties in the proofs.
Acknowledgment. First of all, I am sincerely grateful to Rolf Niedermeier
who gave me the opportunity to work in his group. I am indebted for all
his contributions of time, guidance, and advice. In addition, I would like to
thank all my coauthors and (former) colleagues with whom I have worked
during the last four years. Among them, I am especially grateful to Christian
and Ondřej from whom I have learned much. Last but not least, I would
like to thank my room mate André for countless hours of discussion.
Special thanks goes to my parents and my wife, without them this work
would not have been possible.
13
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J. Guo, S. Hartung, C. Komusiewicz, R. Niedermeier, and J. Uhlmann.
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14
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AAAI Press, 2010 (cited on p. 11).
[Guo+11]
J. Guo, S. Hartung, R. Niedermeier, and O. Suchý. “The parameterized
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J. Guo, S. Hartung, R. Niedermeier, and O. Suchý. “The parameterized
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S. Hartung, C. Komusiewicz, A. Nichterlein, and O. Suchý. “On
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S. Hartung, C. Komusiewicz, and A. Nichterlein. “On structural parameterizations for the 2-Club problem”. In: Proceedings of the 39th
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[HN10]
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parameterized by conservation”. In: Proceedings of the 7th Annual
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15
[HN13a]
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[Jan11]
B. M. P. Jansen. Kernelization for a hierarchy of structural parameters.
Invited talk at the 3rd Workshop on Kernelization (Worker 2011).
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rker/slides/Bart-Jansen.pptx. 2011 (cited on p. 12).
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explanations for 2-d tree- and linearly-ordered data”. In: Proceedings of
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[Mar08]
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16
Contents
1 Introduction and Overview
21
2 Preliminaries and Notation
2.1 Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Classical Complexity Theory . . . . . . . . . . . . . . . . . .
2.3 Parameterized Algorithmics . . . . . . . . . . . . . . . . . . .
2.3.1 Fixed-Parameter Tractability . . . . . . . . . . . . . .
2.3.2 Fixed-Parameter Intractability . . . . . . . . . . . . .
2.3.3 Kernelization . . . . . . . . . . . . . . . . . . . . . . .
2.3.4 Kernelization Lower Bounds . . . . . . . . . . . . . . .
2.3.5 Structural Parameterizations . . . . . . . . . . . . . .
2.3.6 Stronger Parameterizations and Parameter Hierarchies
2.3.7 Running Time Lower Bounds Based on (S)ETH . . .
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3 Metric Dimension on Special Graph Classes
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Related Work . . . . . . . . . . . . . . . . . .
3.1.2 Our Contribution . . . . . . . . . . . . . . . .
3.1.3 Organization and Further Notation for Paths
3.2 Hardness Reduction . . . . . . . . . . . . . . . . . .
3.2.1 General Ideas and Concepts of the Reduction
3.2.2 Formal Description of the Reduction . . . . .
3.2.3 Correctness of the Reduction . . . . . . . . .
3.3 W[2]-Completeness . . . . . . . . . . . . . . . . . . .
3.4 Running Time and Approximation Lower Bounds . .
3.5 Conclusion and Open Questions . . . . . . . . . . . .
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4 Parameterizing 2-Club by Structural Graph Parameters
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Related Work . . . . . . . . . . . . . . . . . .
4.1.2 Further Structural Parameters . . . . . . . .
4.1.3 Our Contribution . . . . . . . . . . . . . . . .
4.1.4 Preliminaries: Twin Classes . . . . . . . . . .
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4.2
4.3
4.4
4.5
4.6
4.7
4.8
NP- and W[1]-Hardness Results . . . . . . . . . . . . . . . . . 83
4.2.1 Clique Cover Number and Domination Number . . . . 83
4.2.2 Distance to Bipartite Graphs . . . . . . . . . . . . . . 86
4.2.3 Average Degree, h-index, and Degeneracy . . . . . . . 88
Fixed-Parameter Tractability Results . . . . . . . . . . . . . . 96
4.3.1 Distance to (Co-)Cluster Graphs and Cographs . . . . 96
4.3.2 Treewidth . . . . . . . . . . . . . . . . . . . . . . . . . 100
Kernelization: Algorithms and Lower Bounds . . . . . . . . . 104
4.4.1 A Quadratic-Vertex Kernel for Cluster Editing Set Size104
4.4.2 A Linear Kernel for Feedback Edge Set Size . . . . . . 107
4.4.3 Lower Bounds with Respect to Bandwidth and Vertex
Cover . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
On the Optimality of the Dual Parameter Algorithm . . . . . 114
Implementation and Experiments . . . . . . . . . . . . . . . . 117
4.6.1 Implemented Algorithms . . . . . . . . . . . . . . . . . 117
4.6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Conclusion and Open Questions . . . . . . . . . . . . . . . . . 124
Tables with Full Experimental Results . . . . . . . . . . . . . 124
4.8.1 Experimental Results for Random Instances . . . . . . 125
4.8.2 Experimental Results for Real-World Instances . . . . 127
5 Parameterizing Local Search for TSP by Neighborhood Measures129
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.1.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . 131
5.1.2 Our Contribution . . . . . . . . . . . . . . . . . . . . . 131
5.2 Preliminaries: Parameterized Local Search and Distance Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5.3 Parameterized Complexity of LocalTSP on General Graphs 137
5.3.1 Running Time Lower Bounds . . . . . . . . . . . . . . 137
5.3.2 Tractability . . . . . . . . . . . . . . . . . . . . . . . . 141
5.4 Parameterized Complexity of LocalTSP on Planar Graphs . 146
5.4.1 Non-Existence of a Polynomial-Size Kernel . . . . . . 146
5.4.2 Fixed-Parameter Tractable Cases . . . . . . . . . . . . 150
5.5 Conclusion and Open Questions . . . . . . . . . . . . . . . . . 152
6 Parameterizing Vector Explanation by Properties of Numeric Vectors
153
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
6.1.1 Related Work . . . . . . . . . . . . . . . . . . . . . . . 155
6.1.2 Our Contribution . . . . . . . . . . . . . . . . . . . . . 155
6.1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . 158
6.2 Further Notation and Combinatorial Properties . . . . . . . . 159
18
6.3
6.4
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Parameterization by Input Smoothness . . . . . . . . . . . . . 164
Parameterizations of the Size and the Structure of Solutions . 171
Conclusion and Open Questions . . . . . . . . . . . . . . . . . 182
7 Conclusion and Outlook
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Bibliography
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19
1 Introduction and Overview
In their landmark papers in the early 1970’s, the Turing award winners
Cook [Coo71] and Karp [Kar72] introduced the fundamental concept of
NP-completeness, leading to the central question in computational complexity theory: Is P equal to NP? In a nutshell, defining the complexity class P
to contain all “efficiently solvable problems”, meaning that there is a solving
algorithm whose computational steps can be upper-bounded by a polynomial
in the input size, Karp [Kar72] proved a list of 21 fundamental problems
as NP-complete, implying that either each or none of them is in P. Unfortunately, after more than four decades the answer whether NP-complete
problems are efficiently solvable is still open. Nevertheless, the introduced
classification tool, the so-called polynomial-time reduction, opened a rich
and fruitful line of research and nowadays thousands of problems arising in
various applications have been identified as NP-complete (see Garey and
Johnson [GJ79] for an early list).
Consider one of the most famous and most intensely studied NP-complete
problems, the Traveling Salesman problem (TSP), where, in its original
formulation, a salesman wants to visit all his customers. Thus, given the
shortest distance between any pair of customers, the problem is to find
a shortest round-tour visiting all customers. Among obvious applications
of TSP in logistic and transport planing tasks, it finds applications in
microchips manufacturing, computer file sequencing, DNA sequencing, etc.,
see Applegate et al. [App+11] and Pop [Pop12] and the references therein.
Due to the frequent occurrence of NP-complete problems in applications,
in the past decades there evolved several approaches to cope with their
computational hardness. Therein, TSP was an often chosen platform for
the introduction of new ideas and algorithmic techniques [App+11]. These
approaches roughly follow one (or a combination) of the following algorithm
design paradigms:
1. Exact Algorithms: Accept a super-polynomial (typically, exponential) running time but still solve the problem exactly [DF99, FK10,
Woe01]. For example, the classical Held/Karp-algorithm [HK62] for
TSP finds a shortest round-tour visiting n customers in O(2n · n2 ) time.
2. Algorithms with Bounded Solution Inaccuracy: Insist on the
polynomial running time but allow the solution to be inaccurate within
21
proven upper bounds. For example, assuming a metric distance function
between customers for TSP, a round-tour which is not worse than a
factor of 1.5 away from the length of an optimal one can be found in
polynomial time [Chr76]. This type of algorithm is called approximation
algorithm and the corresponding field is one of the most active and
most developed fields in algorithm theory [Aus+99, Vaz01, WS11].
3. Heuristic Algorithms: The realm of heuristic algorithms is formed
by those algorithms whose performance and solution quality often have
been empirically explored but there exists no provable guarantee (see
Michalewicz and Fogel [MF04] for a general overview of used techniques
and for theoretical limits see Hemaspaandra and Williams [HW12]).
The wide field of local search algorithms, where starting with some
solution one iteratively tries the improve it by small modifications,
follows this approach. For example, the Lin-Kernighan-heuristic for
TSP [LK73] iteratively improves a round-tour by exchanging up to
three “edges” (a way from one customer to another) until no further improvements can be obtained by this “neighborhood search”.
It belongs to the best performing algorithms for TSP in practical
settings [App+11, JM04].
Besides the three design paradigms described above there are many additional algorithm design techniques or approaches of analysis that form
their own subfield in algorithm theory. We list the most important ones:
There is the approach of average-case analysis where one analyzes the average (or most probable) behavior of an algorithm [BT06] instead of its
worst-case behavior possibly occurring only on a small set of “pathological”
inputs (see Karp [Kar77] for an application to TSP). There is also the
relatively new approach of smoothed analysis [ST09]. Last but not least,
randomized algorithms, in difference to all other approaches, do not proceed
in a deterministic manner but rather by using some source of randomness
and thus their running time and/or solution quality depends on the random
choices [MR95, MU05].
This thesis lies in the field of parameterized algorithmics which can be
seen as a more fine-grained approach following the paradigm of exact algorithms [DF99, FG06, Nie06]. Roughly speaking, parameterized algorithmics
deals with exploiting the (input) structure caused by a bounded parameter
value (e. g. maximum degree in a graph) to design exact solving algorithms
whose unavoidable exponential running time part only depends on the parameter value. For example, a shortest round-tour for TSP through n-points
all lying on the convex hull in the euclidean plane can be found in polynomial
time. However, if there are up to k inner points not lying on the convex
hull, then TSP can be solved in O(2k · k 2 · n) time [Dei+06]. Thus, in
22
applications where only few inner points occur compared to the total number
of points n, this algorithm offers a dramatic performance increase over the
classical Held/Karp-algorithm.
The development of parameterized algorithmics has one of its main motivations from the “unexplained success of heuristic algorithms” [Kar11]
often observed in practical settings when solving NP-hard (a superset of
NP-complete) problems. It is quite tempting to explain this discrepancy
between the “theoretical intractability predicate” of being NP-hard but, nevertheless, allowing for “efficiently” solvable large-scale real-world instances.1
The most plausible and widely accepted explanation of this behavior is
that real-world instances often exhibit a certain structure caused by the
application they arise from and that heuristic algorithms are tuned to the
characteristics of these instances. Being aware of the background application,
some of these structures are obvious whereas other “hidden structures” are
less trivial to identify and are themselves subject of empirical and theoretical
studies. We stress the last point via the following two case studies. Further
examples have been recently discussed at the Symposium on Structure in
Hard Combinatorial Problems [Struc13].
• In the past decades the analysis of complex social networks has emerged
into a general tool for a vast number of scientific disciplines (see Easley
and Kleinberg [EK10] for a more general perspective on networks). For
an example which is closest to our field consider collaboration networks
such as they arise when studying the co-author relationship between
researchers as they are collected by the DBLP database [DBLP] and the
Mathematical Reviews Database (MathSciNet) [MathSci]. When studying these networks, it comes without surprise that the maximum number
of co-authors per author is relatively small compared to the overall
number of authors. However, the small world phenomenon [LH08,
TM69] and also the power law degree distribution [BA99] are examples
of structures that can be found in most social networks and that are
less obvious or even counter-intuitive to the small number of coauthors.
When representing social networks as graphs in a straightforward way,
the small world phenomenon and the power law degree distribution
imply a small average vertex distance and h-index, turning them into
valuable “parameterizations” of social network problems. For the MathSciNet [MathSci] collaboration network, the Erdős number project (see
http://www.oakland.edu/enp/trivia/) collects interesting statisti1 For
example, the Concorde TSP solver (http://www.math.uwaterloo.ca/tsp/
concorde/) finds shortest round-tours for about 86,000 customers. Nowadays, even
cellphones are (typically) able to solve TSP-instances with up to 1000 customers
exactly (http://www.tsp.gatech.edu/iphone/).
23
cal data such as the Erdős number (length of a shortest “chain” of
collaborations to the famous mathematician Paul Erdős) which is on
average 4.65, see Grossman [Gro02] for further information. In other
words the parameter average vertex distance is less than five in this
network.
• The satisfiability problem for boolean formulas, briefly called Sat,
which indeed was the first problem characterized as NP-complete by
Cook’s theorem [Coo71], in recent years turned into a generic “modeling tool” used to solve problems arising in diverse areas such as
hard- and software verification, planning, scheduling, AI planing etc.,
see Gomes et al. [Gom+08, Chapter 2] and the references therein.
The reason for that is the enormous success in the development of
Sat-solvers allowing to solve instances with millions of variables and
constraints (see Biere et al. [Bie+09] for a description of used techniques). When solving Sat-instances it has been observed, and nowadays is known as the phase transition phenomenon, that the hardness
of randomly created instances is heavily influenced by the clause-tovariable ratio α (the number of variables in a formula divided by the
number of constraints/clauses) where the hardest instances for the
special problem variant 3-Sat occur for α ≈ 4.26 [Gom+08]. Besides
these empirical investigations, there has been also a serious interest to
theoretically characterize structures causing the computational hardness. For example, it has been studied to find small so-called backdoors
for Sat-instances, that is, a set of variables such that once the correct
assignment for them has been found, the rest of the formula falls into
a polynomial-time decidable class. Hence, even if a Sat instance is not
entirely contained in an efficiently solvable class, the parameter “size
of backdoors” provides an appealing way to measure its distance from
such a class and thus follows the general approach of “distance from
triviality” parameterization [GHN04]. See Gaspers and Szeider [GS12]
for a survey on backdoors applied to Sat.
The identification of valuable parameters has been even denoted as the
“art of problem parameterization” and developed into a central concern of
parameterized algorithmics [Nie06, Nie10]. These parameters, and the restrictions on the problem structure that are implied by parameters taking small
values, form the basis for the development of algorithms whose “expensive
part” (exponential running time) only depends on the parameter value and
thus admit a good behavior for reasonably small parameter values. As the
term “art” suggests and as already illustrated above with the examples of
social networks and satisfiability testing problems, there is no cookbook on
24
how to find these parameters as, in principle, they could be any quantity
of the input of a computational problem. This motivates the development
of systematic approaches that help to identify reasonable “problem parameterizations” and to chart the borderline between tractable and intractable
cases. Demonstrating how to apply these systematic approaches to NP-hard
problems forms the central concern of this thesis.
Approaches to Identify Tractable Cases of NP-Hard Problems. We
present and conduct three systematic approaches to identify parameterizations or problem restrictions helping to answer the following question: Which
properties of a combinatorial problem determine or at least significantly
influence its computational hardness?
Our first and second approach follow the paradigm of exploring different
restrictions on the structure of the input. In difference to that, our third
approach follows the paradigm of local search and restricts the structure of
the “neighborhood” in which an improved solution is looked for. We will
demonstrate the effectiveness of the approaches by applying them to three
different graph problems. This means that we will classify the computational
complexity of each of these problems with respect to the “parameter space”
derived from the approach. We will further demonstrate the versatile usability of the second approach by applying it to a number problem. The
three approaches are as follows (the applicability of the first is restricted to
graph problems):
1. Restriction to Special Graph Classes: Determine the computational complexity of graph problems that are NP-hard on general
graphs when restricting the input graph to be contained in a special
graph class.
2. Structural Parameterizations: Determine the dependence of the
computational complexity of NP-hard problems when parameterized
by different structural parameters. For example, the diameter and the
h-index of the input graph are naturally occurring structural parameters
for NP-hard graph problems.
3. Local Search with Different Neighborhoods: Explore the impact
of different neighborhood structures on the complexity of the “local
search“-variant of a computational problem. As mentioned above, local
search-based algorithms start with any feasible solution and step by
step try to improve it by searching within the “neighborhood” of the
already known solution. We explore different ways on how to define
these neighborhoods and study the resulting computational complexity.
When applied to graph problems, what is common to the first and second
25
approach is that they aim at measuring the impact of different restrictions
on the structure of the input graph on the computational complexity of
the problem. Many graph classes are defined by a (possibly infinite) set of
forbidden structures. For example, the cograph graph class consists of all
graphs not having the P4 structure (an induced path on four vertices) as an
induced subgraph. Thus the approach of investigating different graph classes
can be viewed as the attempt to identify those structures that determine
the computational hardness of the problem. In difference to that, the
second approach, instead of excluding a certain structure, rather deals with
measuring the size or the number of occurrences of a certain structure in the
input graph and analyzes the influence of this quantity on the hardness of the
problem. For example, having a problem which is polynomial-time solvable
on cographs, in the spirit of the “distance from triviality” approach [Cai03,
GHN04], it is a natural concern to measure the computational hardness of
the problem in terms of the “distance” of the input graph to a cograph.
The third approach, instead of investigating restrictions on the input,
considers restrictions on the “solution search space”. Therein, it follows the
methodology of local search where one is not searching for an overall optimal
solution but basically is satisfied with finding an improvement of an already
known solution. This is accompanied by a restriction on the structure of
the neighborhood of the already known solution in which an improvement is
sought for. Hence, it can be viewed as a restriction on the solution search
space. Similar to the second approach we here try to quantify the impact of
the size of the neighborhood on the hardness of the problem.
What is common to these three approaches is that the corresponding
parameterizations and the implied problem restrictions can be naturally
organized in a hierarchy illustrating the relation from “highly general” down
to more restricted problem variants. We demonstrate that, by organizing the
relationships between different parameterizations in that way, this naturally
leads to a clean and comprehensive picture of problem properties and their
influence on the computational complexity of the problem. Most importantly,
the relations between different parameterizations allow to easily transfer
tractability and intractability results between them. In this way each of the
three approaches yields a “hierarchically-structured” set of parameterizations
and this is what we call a parameter space. Then, we aim at charting the
border between intractability and tractability within these parameter spaces
and thus try to learn which structures in the input or in the solution search
space cause the computational hardness of a problem.
Thesis Overview. This thesis is divided into seven chapters. It starts with
two introductory chapters. This chapter contains an introduction to the
26
scope and the central concepts of this thesis and in Chapter 2 we provide the
relevant notation and concepts from classical and parameterized complexity
theory. In particular, we formalize the notion of structural parameterizations
and discuss its usability on more concrete examples. After having provided
the basic technical concepts we proceed by describing in Chapters 3 to 6 the
main technical results.
Following the first approach to consider special graph classes in order
to identify tractable cases of NP-hard (graph) problems, we consider in
Chapter 3 the computational hardness of the Metric Dimension problem
on bipartite graphs with maximum degree three. The problem is, given a
graph and an integer k, to select at most k vertices such that any two vertices
differ by their distance to at least one of these selected vertices. It was known
that Metric Dimension is NP-complete on bipartite graphs [ELW12] but
the parameterized complexity with respect to k was open even on general
graphs. We prove that Metric Dimension is presumably parameterized
intractable (W[2]-hard) with respect to parameter k even on bipartite graphs
with maximum degree three. In addition, we prove that there cannot be
a polynomial-time approximation algorithm achieving a factor of o(log n)
(unless P = NP).
In Chapter 4 we consider the parameterized complexity of the 2-Club
problem with respect to the parameter space of structural graph parameterizations. 2-Club is the problem is to find a largest subgraph in a given
graph in which each vertex pair is adjacent or has at least one common
neighbor. 2-Club has been considered with respect to several graph classes:
It is trivial on cographs and polynomial-time solvable on bipartite and interval graphs [Sch09]. Following the second approach to identify tractable
cases of NP-hard problems this motivated the study of structural “distance
from triviality” parameterizations [GHN04] such as distance from cographs
and bipartite graphs. We obtain several hardness and tractability results
within the broad parameter space of these structural parameterizations. For
example, we show fixed-parameter tractability with respect to the parameter
distance to cographs and NP-hardness on graphs where deleting one vertex
leads to a bipartite graph. Notably, we also prove a tight lower bound for
an already known parameterized algorithm with respect to the parameter
distance to 2-club. However, we demonstrate by an implementation of this parameterized algorithm that, in discrepancy to this (theoretical) lower bound,
on practical instances it outperforms all previous implementations. Thus,
although we performed a broad study of the parameter space of structural
input parameterizations for 2-Club, this behavior cannot be explained by
the insights we have so far, and thus demonstrates that further study of
(hidden) structures is necessary to explain this.
Our findings for a local search variant of TSP with respect to different
27
neighborhood structures are described in Chapter 5. Studying different neighborhood structures was motivated by Marx [Mar08] who proved hardness for
the so-called edge-neighborhood. According to the third approach to identify
tractable cases of NP-hard problems this result drove us to systematically
explore the structures in this neighborhood which determine the computational hardness and to search for other neighborhood measures admitting
tractability. We further extended this approach by mixing it with the first
approach (restriction to special graph classes), meaning that we investigated
the computational complexity of different neighborhoods on planar graphs.
In our last case study, presented in Chapter 6, we apply the second approach
of structural parameterization of the input to a number problem called
Vector Explanation and thus demonstrate the broad range of applications
of this approach. Vector Explanation is the problem, given an integer
vector, to find a smallest set of homogeneous integer vectors that sum to
the input vector. The reason for applying the second approach of structural
parameterizations to this problem is that by a simple preprocessing step one
can show that the “natural” parameterization by the solution size is within a
factor of two equal to the number of vector entries. This motivates to consider
different parameterizations and, indeed, algorithms for these parameters
have been already developed in order to exploit them for approximation
algorithms [Bie+11, LSY07]. Moreover, considering an application in cancer
treatment therapy, we believe that much of the parameters considered in
our work are reasonably small in instances arising in these applications.
Interestingly, by following this approach we revealed the different behavior
of two problem variants with respect to relevant parameters.
Concluding the thesis, in Chapter 7 we provide a résumé of our findings
and point to interesting open questions and possibilities for follow-up work.
28
2 Preliminaries and Notation
In this chapter we provide the relevant notions from classical and parameterized complexity theory. As in the rest of this work three out of four chapters
discuss graph problems (Chapters 3 to 5), we also introduce the relevant
notions for graphs here.
2.1 Graph Theory
Unless explicitly stated otherwise, all graphs considered in this thesis are
undirected and simple. Each such graph is formally described by a tuple
G = (V, E) with the set of vertices V and a set of edges E ⊆ {{u, v} | u, v ∈
V }. For an edge e = {u, v} the vertices u and v are called the endpoints of e.
Moreover, if the graph is clear from the context, we use n = |V | and m = |E|.
Let G = (V, E) be a graph. We set V (G) = V and E(G) = E. A
graph G0 = (V 0 , E 0 ) is a subgraph of G if V 0 ⊆ V and E 0 ⊆ E. For a
vertex subset S ⊆ V , denote by G[S] the subgraph induced by S, that is,
G[S] = (S, {{u, v} ∈ E | u, v ∈ S}). In addition, we briefly write G − S for
G[V \ S]. For a vertex v ∈ V , denote by N (v) = {u ∈ V | {u, v} ⊆ E} the
neighborhood of v and by N [v] = N (v) ∪ {v} the closed neighborhood of v.
Let deg(v) = |N (v)| be the degree of v. We say that v is adjacent to all
vertices in N (v) and it is incident to each edge {u, v} ∈ E. The complement
graph of G is G = (V, {{u, v} ∈
/ E | u, v ∈ V }).
A path from a vertex u to a vertex v of length t − 1 is a sequence
(v1 , v2 , . . . , vt ) of vertices such that (i) v1 = u and vt = v, (ii) {vj , vj+1 } ∈ E
for all 1 ≤ j < t, and (iii) it is called simple if vj 6= vl for all 1 ≤ j < l ≤ t.
If not explicitly stated otherwise, then a path is required to be simple. A sequence (v1 , v2 , . . . , vt , v1 ) where (v1 , . . . , vt ) is a path, t > 2, and {v1 , vt } ∈ E
is a cycle of length t. A connected component of a graph is a maximal subset
of vertices that pairwisely have a path between them. This naturally defines
a partition of a graph into its connected components and we say that vertices
belonging to the same connected component are connected and otherwise
they are disconnected. A graph is connected if it consists only of one connected component and otherwise it is disconnected. For two vertices u and v
we use dist(u, v) to denote the distance of u and v, that is, the length of a
shortest path between u and v (especially, dist(v, v) = 0). If u and v are
29
disconnected, then dist(u, v) = ∞. For a vertex v ∈ V and an integer t ≥ 1,
denote by Nt (v) = {u ∈ V \ {v} | dist(u, v) ≤ t} the set of vertices within
distance at most t to v. Moreover, we set Nt [v] = Nt (v) ∪ {v}. If the graph is
not clear from the context, then we will add a super- or subscript indicating
the corresponding graph, for example, we write NG (v) to emphasize that we
are speaking about v’s neighborhood in G.
Graph Measurements and Graph Classes. We introduce some prominent
graph measurements and graph classes.
Let G = (V, E) be a graph. We define the following measurements and
graph features.
• The maximum degree ∆ of G is the size of the largest neighborhood of
a vertex in G, formally ∆ = maxv∈V deg(v).
P
• The average degree is |V1 | · v∈V deg(v).
• The diameter is the length of a shortest path between any two vertices,
formally maxu,v∈V dist(u, v).
• The domination number of a graph is the minimum size of a dominating
set, that is, a vertex set such that each vertex is contained in it or has
at least one neighbor in it.
• A clique is a vertex set in which each pair of vertices is adjacent.
• A vertex cover is a vertex set such that each edge has at least one
endpoint in it.
• An independent set is the (vertex-)complement of a vertex cover.
• A set of edge insertions and deletions is a cluster editing set if it
transforms G into a graph whose connected components are all cliques,
that is, a so-called cluster graph (see below).
• A set of vertices/edges is a feedback vertex/edge set if its deletion
results in a graph without any cycle.
• The degeneracy of a graph is the smallest number d such that each
subgraph has at least one vertex of degree at most d.
• A graph has h-index k if k is the largest number such that the graph
has at least k vertices of degree at least k [ES12].
• A k-coloring of G for some k ∈ N is a function c : V → {1, . . . , k}
which is called proper if c(v) 6= c(w) for all {v, w} ∈ E. The chromatic
number of G is the smallest k such that G admits a proper k-coloring.
We define the following graph classes, that is, subsets of the set of all
graphs (see Brandstädt et al. [BLS99] for a more comprehensive list):
30
• A graph is a forest if it does not contain any cycle and it is a tree if,
additionally, it is connected.
• We denote by Pt an induced path on t vertices. The class of Pt -free
graphs consists of all graphs not containing any induced Pt . The
P4 -free graphs are called cographs, P3 -free graphs are so-called cluster
graphs, and connected P3 -free graphs are cliques. Equivalently, a graph
is a clique if each vertex pair is adjacent. A graph is a co-cluster graph
if its complement graph is a cluster graph.
• A graph is planar if it can be embedded/drawn in the plane without
crossing edges (refer to Brandstädt et al. [BLS99, Section 7.3] for a
more formal definition).
• The set of k-partite graphs consists of all graphs whose vertex set can
be partitioned into k sets such that each partition is an independent
set. Equivalently, a graph is k-partite if it admits a proper k-coloring.
In the special case of k = 2 we speak about bipartite graphs.
2.2 Classical Complexity Theory: Problems,
NP-Completeness, and Running Time
Analysis
In this section we provide a compact introduction to the relevant concepts
of “classical” complexity theory.
Decision Problems. We typically consider decision problems and, although
we will quickly move on to a more abstract view, we first give a definition in
terms of formal language theory.
Definition 2.2.1 (Decision Problem). For a fixed alphabet Σ denote by Σ∗
the set of all finite words consisting of letters from Σ. A set L ⊆ Σ∗ is called
language and the corresponding decision problem is given a word x ∈ Σ∗ ,
decide whether x ∈ L or not. An instance x ∈ Σ∗ is a yes-instance if x ∈ L
and otherwise x is a no-instance.
As the choice of the alphabet Σ (one can always think about Σ = {0, 1})
and the precise way of encoding an instance is not important, in the following
we will prefer to formulate decision problems in natural language. We give
an example by defining the NP-complete Vertex Cover problem:
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Vertex Cover [GJ79](GT1)
Input: An undirected graph G = (V, E) and a positive integer k.
Question: Is there a size at most k vertex cover in G?
To match Definition 2.2.1 one can define the “Vertex Cover-language”
as follows, which, however, still abstracts from the way on how to encode
graphs, numbers, etc.:
LVC = {(G, k)| G has a a vertex cover of size at most k}.
NP-Completeness. We introduce the central concept of polynomial-time
reductions leading to the definition of NP-hard problems.
Definition 2.2.2 (Polynomial-Time Reductions). A problem A ⊆ Σ∗ is
(many-to-one) polynomial-time reducible to another problem B ⊆ Σ∗ if
there is a function f : Σ∗ → Σ∗ such that for any instance I ⊆ Σ∗ it holds
that (i) I ∈ A ⇔ f (I) ∈ B and (ii) there is an algorithm computing f in
polynomial-time in the input size.
Remark. Polynomial-time reductions (Definition 2.2.2) are sometimes called
Karp reductions to emphasize the difference to concepts such as Turing
reductions where one is allowed to reduce one instance into many instances.
Throughout this thesis the term polynomial-time reduction will always refer
to Definition 2.2.2.
A problem is called to be NP-hard if all problems in NP are polynomialtime reducible to it. By the famous “Cook theorem” [Coo71] the so-called Sat
problem has been classified, for the first time, as an NP-hard problem. The
central consequence is that solving an NP-hard problem in polynomial-time
would imply P = NP. NP-hard problems in NP are called NP-complete. We
refer to Garey and Johnson [GJ79, Chapters 1-2] for a more comprehensive
introduction to the theory of NP-completeness, concerns about encodings,
and a formal definition of the (non-)deterministic Turing machine model
leading to a formal definition of NP and P.
Running Time Analysis: Machine Model and “bigO”-Notation. We will
always analyze running times with respect to the unit-cost random-access
machine (RAM) model. Most important to this is that accessing any “memory
cell” and performing basic operations such as simple arithmetics on numbers,
assignments, if-statements, etc. only count as one computational step which
can be performed in constant time.
We also employ the “bigO”-notation for an asymptotic running time analysis
of algorithms. Its main purpose is to suppress constant factors and loworder terms. Formally, for some function f : N → N an algorithm for a
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problem is running in O(f ) time if there is a constant c and an integer n0
such that the algorithm needs on all input instances of size n ≥ n0 at
most c · f (n) computational steps. See Cormen et al. [Cor+01] for a more
comprehensive introduction.
2.3 Parameterized Algorithmics
This section provides a brief introduction to parameterized algorithms and
complexity. We list the basic definitions and notations. Refer to the monographs of Downey and Fellows [DF99], Flum and Grohe [FG06], and Niedermeier [Nie06] for a more comprehensive introduction.
2.3.1 Fixed-Parameter Tractability
All problems that we consider are NP-hard and thus, assuming P 6= NP, resist
any attempt to design an algorithm solving them in polynomial-time. In the
“classical” framework time complexity is measured by a function in the input
size, for example the number of bits needed to encode the tuple (G, k) in a
Vertex Cover instance. The central concern of parameterized complexity
is to provide a “more fine-grained” view by measuring the computational
complexity of a problem not only in terms of its “encoding size” but rather
it tries to identify certain problem-specific parameters and to analyze their
impact to the complexity of the problem. For example, Vertex Cover with
respect to the standard parameter “solution size” k is fixed-parameter tractable
as there is a simple branching-based algorithm solving any instance (G, k)
in O(2k · |G|) time [DF99]. From this point of view, one could say that
the computational complexity of Vertex Cover is rather determined by
parameter k than by the size of the graph |G|, as it contributes only a linear
factor to the running time. We first formalize the term parameterized problem.
Definition 2.3.1 (Parameterized Problem). For an alphabet Σ, a parameterized problem is a language L ⊆ Σ∗ × N. In any instance (I, k) ∈ Σ∗ × N we
refer to I ⊆ Σ∗ as the classical problem and k is denoted as the parameter.
Remark. Flum and Grohe [FG06] define a parameterized problem slightly
different, as in their definition the parameter k has to be computable in
polynomial time from the classical problem instance I. We adopt the notation
of Downey and Fellows [DF99]. We will discuss this issue when defining
structural parameterizations (see Section 2.3.5).
We now define the central notion of parameterized algorithmics.
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Definition 2.3.2 (Fixed-Parameter Tractability). A parameterized problem
L ⊆ Σ∗ × N is fixed-parameter tractable and contained in the complexity
class FPT if there is a computable function f : N → N, a constant c, and an
algorithm that decides each instance (I, k) ∈ Σ∗ × N in f (k) · |I|c time.
Remark. Following the notation of Downey and Fellows [DF99], Definition 2.3.2 describes the class of strongly uniform fixed-parameter tractable
problems.
It is important to emphasize that the function f in Definition 2.3.2 solely
depends on the parameter k and thus the corresponding parameterized
algorithm runs in polynomial time for constant parameter values k and the
degree of the polynomial in the input size is independent from the parameter.
In difference to that, a parameterized problem that is polynomial-time
solvable for constant parameter values is contained in XP:
Definition 2.3.3 (Complexity Class XP). A parameterized problem L ⊆
Σ∗ × N is contained in XP if there is a computable function f : N → N and
an algorithm deciding each instance (I, k) ∈ Σ∗ × N in |I|f (k) + f (k) time.
A recent development extends parameterized complexity analysis into
a multivariate complexity analysis where multiple parameters are combined [FJR13, Nie10].
2.3.2 Fixed-Parameter Intractability
Analogously to the concept of NP-hardness, there is a theory of parameterized
intractability. The two basic classes of parameterized intractability are W[1]
and W[2]. A parameterized problem shown to be W[1]- or W[2]-hard by
means of a parameterized reduction from another W[1]- or W[2]-hard problem
is believed to be not fixed-parameter tractable since otherwise FPT = W[1]
or FPT = W[2], respectively.
Definition 2.3.4 (Parameterized Reduction). A parameterized reduction
from a parameterized problem L to another parameterized problem L0 is a
function r such that for any instance (I, k) for L with r(I, k) = (I 0 , k 0 ) it
holds that
(i) r(I, k) is computable in O(f (k) · |I|c ) time for some computable function f and a constant c,
(ii) k 0 ≤ g(k) for some computable function g, and
(iii) (I, k) ∈ L ⇔ (I 0 , k 0 ) ∈ L0 .
Downey and Fellows [DF99] established the W -hierarchy of presumably
parameterized intractable classes lying between FPT and XP:
FPT ⊆ W[1] ⊆ W[2] ⊆ W[3] ⊆ . . . ⊆ W[t] ⊆ W[P ] ⊆ XP
34
Its is unconditionally proven that FPT 6= XP [DF99] and the standard
working hypothesis, similar to P 6= NP, is that all inclusions above are
strict, implying FPT 6= W[i] for all i ≥ 1. As mentioned above, Vertex
Cover parameterized by the solution size k is contained in FPT whereas
the NP-complete Independent Set problem parameterized by the solution
size is W[1]-complete [DF99] and thus contained in FPT if and only if
FPT = W[1]:
Independent Set [GJ79](GT20)
Input: An undirected graph G = (V, E) and a positive integer k.
Question: Is there an independent set of size k in G?
Since Vertex Cover is contained in FPT, unless FPT = W[1], there is
no parameterized reduction from Independent Set to Vertex Cover.
However, there is a simple polynomial-time reduction as a graph G = (V, E)
has a vertex cover of size k if and only if it has an independent set of
size |V | − k.
As the (vertex) complement of an independent set is a vertex cover and
vice versa, Vertex Cover can be viewed as the “vertex complement”
of Independent Set, however, the reduction introduced above is not
a parameterized reduction. Though the “graph complement” problem of
Independent Set, the so-called Clique problem, admits a parameterized
reduction from Independent Set to it and thus is also W[1]-hard.
Clique [GJ79](GT19)
Input: An undirected graph G = (V, E) and a positive integer k.
Question: Is there a clique of size k in G?
An independent set in a graph G is a clique in the complement graph G
and vice versa. Hence, computing the complement graph and taking the same
solution-size parameter k is a trivial parameterized reduction from Clique
to Independent Set and vice versa.
2.3.3 Kernelization
The notion of fixed-parameter tractability is closely related to the field
of efficient and effective preprocessing algorithms, called kernelization in
parameterized algorithmics. The surveys of Bodlaender [Bod09], Guo and
Niedermeier [GN07], and Lokshtanov et al. [LMS12] provide an overview
about this rapidly growing subfield of parameterized algorithmics.
Definition 2.3.5 (Kernelization). Let L ⊆ Σ∗ × N be a parameterized
problem and let g : N → N be a computable function. A kernelization
35
algorithm for L of size g reduces each instance (I, k) ∈ Σ∗ × N for L in time
polynomially bounded in |I| + k, to an equivalent instance (I 0 , k 0 ) ∈ Σ∗ × N
such that |I 0 |, k 0 ≤ g(k) and (I 0 , k 0 ) ∈ L ⇔ (I, k) ∈ L. The instance (I 0 , k 0 )
is called kernel of size g and in the special case of g being a polynomial it is
a polynomial kernel.
The following theorem reveals the relation between problems in FPT and
those admitting a kernelization algorithm.
Theorem 2.3.6 ([Cai+97]). A decidable parameterized problem is contained
in FPT if and only if it admits a kernelization algorithm of any size.
A kernel is often described by the exhaustive application of several so-called
data reduction rules, that are, polynomial-time algorithms that transform an
instance (I, k) into an equivalent instance (I 0 , k 0 ). A reduction rule has been
applied exhaustively if applying it once more would not change the instance
and we call these instances reduced with respect to the reduction rule.
For example, the exhaustive application of the well-known Buss reduction
yields an O(k 2 )-size problem kernel for Vertex Cover [BG93, DF99]:
Reduction Rule 2.3.7. For any Vertex Cover-instance (G, k) where G
contains a vertex v of degree either zero or at least k + 1, remove v and all
incident edges from G and decrease k by one.
Observe that after having exhaustively applied Reduction Rule 2.3.7, if
there is a vertex cover of size at most k, then there can be at most k 2 edges
“covered” by these vertices. Hence the resulting instance has at most k + k 2
vertices and at most k 2 edges or it is no-instance for Vertex Cover.
Turing Kernelization. We next describe a prominent relaxation of the
“kernelization concept”, called Turing kernelization, that shares most of its
benefits such as the existence of a well-performing brute-force algorithm
in case of “small” (polynomial) kernels. Notably, there are problems not
admitting a polynomial kernel (see Section 2.3.4 for a classification tool
therefore) but there is even a linear-size Turing kernel for them.
We introduce Turing kernelization via an application to the Clique
problem. Clearly, for a non-trivial Clique-instance (G = (V, E), k) it holds
that the maximum degree ∆ of G is at least k − 1. A straightforward
application of the framework described in Section 2.3.4 proves that Clique
parameterized by ∆ does not admit a polynomial kernel. However, there
is a simple preprocessing step that dramatically reduces the overall graph
size, which needs to be handled by an algorithm solving Clique, in case
of a small maximum degree compared to the number of vertices: For each
vertex v ∈ V solve the Clique-instance (G[N (v)], k − 1). As a clique C is
36
completely contained in N (v) for each v ∈ C, at least one of these instances
is a yes-instance if only if (G, k) is a yes-instance.
The crucial point in the above algorithm is that, measured in terms of
the parameter ∆, each of the instances (G[N (v), k − 1]) is “small”, indeed
G[N (v)] has at most ∆ vertices. The central difference between a “classical”
kernelization and Turing kernelization is that, in the later, it is allowed to
“kernelize” to polynomially-many instances such that the size of each of them
is bounded in the parameter value. Thus, analogously to Theorem 4.4.14,
a Turing kernelization implies the existence of a parameterized algorithm.
We next provide a formal definition of Turing kernelization which is due to
Binkele-Raible et al. [Bin+12] and Lokshtanov [Lok09b].
Definition 2.3.8 (Turing Kernelization). Let L be a parameterized problem
and for some computable function g let a g-oracle be an algorithm deciding
any instance (I, k) of L with |I|, k ≤ g(k) in constant time. A Turing
kernelization of size g is an algorithm with access to a g-oracle deciding each
instance (I, k) of L in time polynomial in |I| + k.
Remark. The bounded-size instances that have been decided by the oracle
during a run of a Turing kernelization are called Turing kernels.
Intuitively, a Turing kernelization is allowed to perform polynomially
many rounds, each consisting of constructing in polynomial time a size at
most g(k) instance (potentially, this depends on the results obtained in
previous rounds) and deciding it in constant time by the help of the oracle.
However, to the best of our knowledge, currently no Turing kernelization
makes actually use of a “non-trivial” combination of the oracle answers. For
example, the above described Turing kernelization for Clique parameterized
by the maximum degree ∆ does not need the “power” of a polynomial-time
algorithm to combine the results of the oracle answers and also the next
oracle question does not depend on the previous oracle answers. Indeed,
all oracle questions could be computed in advance completely independent
from each other and then the outcome of all just needs to be combined by a
logical OR. The observation that the notion of Turing kernelization could
be further constrained without restricting known applications, led to the
introduction of the “truthtable kernelization”-concept [Wel12].
2.3.4 Kernelization Lower Bounds Based on
(Cross-)Compositions
The proof of Theorem 2.3.6 shows that, having classified a problem as fixedparameter tractable by a parameterized algorithm running in f (k) · |I|O(1)
time, immediately implies the existence of a kernel of size f (k). However,
37
due to the NP-hard nature of most of the problems under consideration, f is
typically not polynomial. Hence, having classified a parameterized problem
to be contained in FPT naturally leads to the following two “classical races”
in parameterized algorithmics: (i) find the “smallest” function f admitting a
corresponding parameterized algorithm, and (ii) find the “smallest” function g
admitting a kernelization algorithm of size g.
In this section we introduce a popular framework, mainly consisting of
so-called cross OR/And-compositions, allowing to rule out the existence of
polynomial kernels. The framework builds upon the OR/AND-distillation
conjecture which (informally) states that there is no polynomial-time algorithm that takes as input arbitrarily many instances of an NP-hard problem
and outputs one instance of another problem that is a yes-instance if and
only if at least one of the given instances is a yes-instance (OR-distillation)
or if all of the given instances are yes-instances (AND-distillation). Under
the assumption that NP ⊆ coNP/poly does not hold, the OR-distillation
conjecture has been proven by Fortnow and Santhanam [FS11] and the ANDdistillation by Drucker [Dru12]. Based on this Bodlaender et al. [BJK11]
established the framework of cross OR/AND-compositions. Note that it is
indeed an extension of the “composition”-framework introduced by Bodlaender et al. [Bod+09]:
The framework will allow to consider only instances that are “related”.
Definition 2.3.9 (Polynomial Equivalence Relation). An equivalence relation R ⊆ Σ∗ ×Σ∗ is a polynomial equivalence relation if (i) there is a polynomial-time algorithm that, given any two x, y ∈ Σ∗ , decides whether (x, y) ∈ R,
and (ii) there is a polynomial p such that for each finite set S ⊆ Σ∗ the
number of equivalence classes of S is at most p(n) with n = maxx∈S |x|.
Polynomial equivalence relations provide a handy tool which, for example,
allows to assume that all graphs in a set of instances have the same number
of vertices and edges. Definition 2.3.9 allows to choose R as the equivalence
relation with only one equivalence class and thus “erasing” it from the
following definition and this will be the case in all of our applications of the
framework if not explicitly stated otherwise.
Definition 2.3.10 (AND/OR-Cross-Composition). A language L ⊆ Σ∗
AND/OR-cross-composes into a parameterized problem L0 ⊆ Σ∗ × N with
respect to some equivalence relation R if there is an algorithm that, given t
instances I1 , . . . , It of L all belonging to thePsame equivalence class of R,
t
computes in time polynomially bounded in i=1 |Ii | an instance (I 0 , k 0 ) ⊆
∗
Σ × N such that
(i) parameter k 0 is polynomially bounded in max1≤i≤n |Ii | + log t, and
either
38
(ii) (OR) (I 0 , k 0 ) ∈ L0 ⇔ ∃i ∈ {1, . . . , t} : Ii ∈ L, or
(iii) (AND) (I 0 , k 0 ) ∈ L0 ⇔ ∀i ∈ {1, . . . , t} : Ii ∈ L.
The results of Bodlaender et al. [BJK11], Drucker [Dru12], and Fortnow
and Santhanam [FS11] establish the following:
Theorem 2.3.11. If an NP-hard language L AND/OR-cross-composes into
a parameterized problem L0 , then L0 does not have a polynomial kernel unless
NP ⊆ coNP/poly.
Remark. The condition NP * coNP/poly in Theorem 2.3.11 strengthens
the assumption that NP 6= coNP. It deals with advice Turing machines
and basically states that even if a coNP-Turing machine is equipped with a
polynomial advice (for all length-n inputs it gets the same string/advice of
length nO(1) ), then it is still not powerful enough to decide all NP-languages.
Yap [Yap83] proved that if NP ⊆ coNP/poly, then the polynomial hierarchy
collapses to its third level.
The definition of parameterized reductions (Definition 2.3.4) has been
strengthened to being able to transform lower bounds on the existence of
polynomial kernels from one problem to another [BTY11]:
Definition 2.3.12 (Polynomial-Time and Parameter Transformation). A
polynomial-time and parameter transformation from a parameterized problem L into another parameterized problem L0 is a function r such that for
any instance (I, k) for L with r(I, k) = (I 0 , k 0 ) it holds that
(i) r(I, k) is computable in polynomially-many steps in |I| + k,
(ii) k 0 is polynomially bounded in k, and
(iii) (I, k) ∈ L ⇔ (I 0 , k 0 ) ∈ L0 .
If in Definition 2.3.12 property (ii) can be strengthened to a linear function in k which upper-bounds k 0 , then we call the transformation a linearparameter transformation.
Theorem 2.3.13 ([BTY11]). Let L and L0 be two parameterized problems
such that the classical problem variant of L is NP-complete and the classical
problem variant of L0 is contained in NP. If there is a polynomial-time
and parameter transformation from L to L0 and L0 has a polynomial kernel,
then L has a polynomial kernel.
Remark. As a technical constraint in Theorem 2.3.13, one actually has
to require that L remains NP-hard if the parameter is encoded over a
unary alphabet.
39
2.3.5 Structural Parameterizations
We considered so far only the “standard parameterization” by the solution size
and thus already a classical minimization or maximization problem instance
contains the parameter of interest as a number in the input. However,
the parameter under consideration could be in principle any quantity. In
this section we reveal the difficulties when considering so-called “structural
parameters of the input”. Before that, we remark that there is no formal
definition of what is considered to be a structural parameterization. The
intuitive meaning is that it is a quantitative measurement of some structure
in the input that is rather independent of the problem. For example, any
graph problem can be parameterized by the maximum degree, treewidth,
the size of a feedback edge set, etc. of the input graph.
Consider parameterizing Vertex Cover by the parameter size of an
independent set. We briefly call it Vertex Cover-IS and the corresponding
computational task is to decide for a graph G and two integers k and `
where G has a size-` independent set whether G has a size at most k
vertex cover. According to Definition 2.3.2 any instance of Vertex CoverIS consists of (G, k, `) where (G, k) is the classical problem and ` is the
parameter. Despite the question whether this parameterization provides any
insights for Vertex Cover, this formulation causes some inconsistencies
with its intended meaning, as for a graph G = (V, E) with a size-k vertex
cover the instance (G, k, |V |) is a no-instance for Vertex Cover-IS in case
of |E| > 0. Hence, the difficulty arises not only to check whether G has a
size-k vertex cover but also to verify that there is a size-` independent set
in G which is itself an NP-hard problem.
Remark. Note that, unless P = NP, in the more rigorous definition of FPT
by Flum and Grohe [FG06] one cannot express parameterizations by an
independent set or by the treewidth of the input graph.
Generally, we would like to separate the computational hardness of finding
a witness for the structural parameterization (e. g. a size-` independent set)
from solving the actual problem assuming a correct witness is provided. To
overcome the above problem, throughout this thesis we will thus assume that
(i) there is polynomial-time algorithm computing the structural parameter
from the classical problem instance (corresponds to [FG06]), or
(ii) the instance is extended by a witness such that there is a polynomialtime algorithm able to verify the structural parameter.
Furthermore, in our running time analysis for algorithms solving a parameterized problem, we will neglect the additional term caused by the
polynomial-time algorithm in either case and thus “concentrate” on the
40
algorithm part solving the problem. Indeed, as the computational problem
corresponding to our structural parameters are all contained in NP there is
always a “short” (polynomially in the input length) witness which can be
verified in polynomial-time.
Remark. According to Definition 2.3.2 the parameter will be always an
integer such as the size of the desired vertex cover or, as discussed above,
the size of some structure such as the size of an independent set etc. In the
following when speaking about a structural parameterization we will call
the structure a parameter while actually meaning the size of the structure.
For example, instead of parameterizing by the size of a vertex cover we will
briefly call it a parameterization by a vertex cover.
2.3.6 Stronger Parameterizations and Parameter
Hierarchies
As mentioned above, after having classified a problem to be fixed-parameter
tractable, the “race” starts for the best parameterized algorithm, that is,
those with the modest function f admitting an f (k) · nO(1) -time algorithm.
This is usually accompanied by a race for the smallest polynomial kernel, in
case of its existence. There are many success stories on these developments
that led to fast algorithms and often introduced new techniques, for example,
see the problems Cluster Editing [BB13, CM12] and Feedback Vertex
Set [CCL10, Cyg+11].
However, in recent years the question about tractability with respect to
non-standard parameters, as the structural parameterization discussed in
Section 2.3.5, gained more attention. Strongly motivated by the viewpoint
of multivariate algorithmics [FJR13, Nie10], the task to find the “smallest”
parameter admitting a parameterized algorithm or even a polynomial kernel
arises. As a motivating example, note that there is a 2k-vertex kernel of
size O(k 2 ) for Vertex Cover [CKJ01, NT75] and this is essentially tight as
for all > 0 there is no O(k 2− )-size kernel unless NP ⊆ coNP/poly [DvM10].
Hence the end of the story of Vertex Cover-kernelization seems to be
reached. However, Vertex Cover admits a kernel where the number of
vertices is upper bounded by a cubic term in the size of a feedback vertex
set [JB13]. As every vertex cover is also a feedback vertex set, on all graphs
the parameter minimum feedback vertex set is provably smaller than the
size of a vertex cover and the “gap” can be arbitrarily large, for example, on
trees. This is what called to be a stronger parameterization [KN12].
Definition 2.3.14 (Stronger Parameterization). For a classical problem L ⊆
Σ∗ , let α, β : Σ∗ → N be two functions called parameterizations. Parameter α
is called a stronger parameterization than β if there is a computable function
41
f : N → N such that α(I) ≤ f (β(I)) for all instances I ∈ Σ∗ . If α is a stronger
parameterization than β, then we will call β a weaker parameterization.
Komusiewicz and Niedermeier [KN12] required function f in Definition 2.3.14 to be a linear function whereas Weller [Wel12] required f to
be a polynomial. The main advantage of these more restrictive definitions
is that a polynomial kernel for a parameter k which is stronger than a
parameter k 0 also implies a polynomial kernel for k 0 . Although our definition
(which equals that of Jansen [Jan13]) “only” allows to transfer parameterized
algorithms from stronger parameters to weaker parameters, its advantage is
to being able to cover “exponential” relations between parameters such as
clique-width and rank-width [Oum08].
Remark. Note that the computational complexity of computing the parameters for an instance is completely neglected in Definition 2.3.14. In
addition, the parameterization will always refer to the value of an optimal
solution of the corresponding problem. For example, when speaking about
the parameterization by a feedback vertex/edge edge set, in the context of
Definition 2.3.14 we always refer to a minimum-size feedback vertex/edge set
and thus feedback vertex set is a stronger parameterization than feedback
edge set. Additionally, note that a minimum-size feedback edge set can
be computed in polynomial time via the (edge-)complement of a spanning
tree, whereas finding a minimum-size feedback vertex set is an NP-complete
problem [GJ79](GT7).
Stronger parameterizations directly imply consequences for the existence of
parameterized algorithms for the corresponding parameters. Having a parameterization α that is stronger than β, then all positive results (parameterized
or XP-algorithms) for α do also hold for β and all (W[1]-, W[2]-)hardness
results for β do also hold for α. This situation is similar to the well-known
Hasse diagrams of the inclusion relation of several graph classes, see for
example [BLS99]. Hardness results for special graph classes transfer “up to”
more general graph classes whereas tractability results transfer from more
general classes “down to” to more restricted graph classes.
As for graph classes, it is quite convenient to represent the relations of
different parameterizations with respect to being stronger in a Hasse diagram.
See Figure 2.1 for an example of prominent graph parameterizations (formally
defined in Section 2.1) and their relations. Similar hierarchies can be found
in Jansen [Jan13], Komusiewicz and Niedermeier [KN12], and Weller [Wel12].
See also Jansen [Jan13], Sasák [Sas10], and Sorge and Weller [SW13] for
a detailed explanation of these relations. Their work also contain proofs
showing that certain parameters are unrelated (with respect of being a
stronger parameterization). However, as a diagram showing (stronger)
relations together with the non-existence of them for all pairs of parameters
42
distance
to clique
vertex cover
cluster editing
max leaf
number
maximum independent set
distance to
co-cluster
distance to
cluster
distance to
disjoint paths
feedback
edge set
bandwidth
domination
number
distance to
cograph
distance to
interval
feedback
vertex set
pathwidth
maximum
degree
distance to
chordal
distance to
bipartite
treewidth
h-index
diameter
average vertex
distance
distance to
perfect
degeneracy
Figure 2.1: Hasse diagram of the stronger relationship of graph parameters. An
edge from a higher-drawn parameter β down to a parameter α, indicates that α is
a stronger parameterization than β. For a graph class Π the parameter distance
to Π is the number of vertices that have to be deleted to transform the graph into
one that is isomorphic to a graph in Π.
quickly becomes unstructured and because “non-relations” are not helpful
for our main purpose to transfer (in-)tractability result, Figure 2.1 shows
only the known relations.
2.3.7 Running Time Lower Bounds Based on (S)ETH
Impagliazzo et al. [IPZ01] introduced the (Strong) Exponential Time Hypothesis which states lower bounds on how fast the k-Sat problem can
be solved. Assuming that these hypotheses are true, many lower bounds
for several different problems have been established in recent years. See
Lokshtanov et al. [LMS11] for a survey.
For a constant integer k ≥ 2 the k-Sat problem is defined as follows.
k-Sat
Input: An n-variable boolean formula F in conjunctive normal form
(CNF) with at most k literals in each clause.
Question: Is there an assignment to the variables in F that evaluates
to true?
The general Sat problem [GJ79](LO1) is defined in the same way as k-Sat
except that there is no bound on the number of literals in the clauses of the
input formula.
43
Impagliazzo et al. [IPZ01] introduced the following hypothesis stating a
running time lower bound for 3-Sat.
Hypothesis 2.3.15 (Exponential Time Hypothesis (ETH)). There is a
δ ∈ (0, 1) such that solving 3-Sat requires at least 2δn · |F |O(1) time.
In particular, the ETH implies that the trivial algorithm for Sat that
tests all assignments and runs in O(2n · |F |) time cannot be improved–
not even in case of 3-Sat–to an algorithm running in 2o(n) · |F |O(1) time.
Impagliazzo et al. [IPZ01] also provide the so-called Sparsification lemma that
indeed allows to infer that, unless the ETH fails, there is no 2o(m) · |F|O(1)
time algorithm for 3-Sat where m denotes the number of clauses.
Lower bounds based on the ETH are then usually established as shown
by the following theorem. Therein, for a function p we denote by p−1 the
inverse function.
Theorem 2.3.16. Let L be a parameterized problem such that there is a
polynomial-time and parameter transformation from 3-Sat parameterized by
the number of clauses m to L and let p be the polynomial that bounds the
parameter in any transformed instance of L. Then there is no parameterized
−1
algorithm for L running in 2o(p (k)) · |I|O(1) time for all instances (I, k)
unless the ETH fails.
For example, unless the ETH fails, there is no f (k) · no(k) time algorithm [Che+05] and also no 2o(n) -time algorithm for Independent Set on
n-vertex graphs [LMS11].
The Strong Exponential Time Hypothesis formalizes the belief that for
general Sat one cannot do any better than the above mentioned trivial
algorithm.
Hypothesis 2.3.17 (Strong Exponential Time Hypothesis (SETH)). For
any δ ∈ (0, 1) there exists a k such that solving k-Sat requires at least
2δn · |F |O(1) time.
The SETH negates the existence of an algorithm running in (2 − )n ·
|F|O(1) time for Sat. Note that there is no “Sparsification lemma”-equivalent
for SETH and thus to get an SETH-equivalent of Theorem 2.3.16 one
has to provide a polynomial-time and parameter transformation from Sat
parameterized by the number of variables n.
44
3 Metric Dimension on Special
Graph Classes
The NP-hard Metric Dimension problem is to decide for a given undirected
graph G and a positive integer k whether there is a vertex subset of size at
most k that separates all vertex pairs in G. Herein, a vertex v separates a
pair {u, w} if the distance (length of a shortest path) between v and u is
different from the distance between v and w. We give a polynomial-time
computable reduction from the Bipartite Dominating Set problem to
Metric Dimension on bipartite graphs with maximum degree three such
that there is a one-to-one correspondence between the solution sets of both
problems. There are two main consequences of this.
First, it proves that Metric Dimension on bipartite graphs with maximum degree three is W[2]-hard with respect to the parameter k. This answers
an open question concerning the parameterized complexity of Metric Dimension posed by Lokshtanov [Lok09a, Dagstuhl seminar ’09] and also by
Díaz et al. [Día+12]. In addition, it implies that a trivial nO(k) -time algorithm
cannot be improved to an no(k) -time algorithm, unless FPT = W[1]. Second,
it follows that Metric Dimension on bipartite graphs with maximum degree
three is inapproximable by a factor of o(log n), unless P = NP. This strengthens the result of Hauptmann et al. [HSV12] who proved APX-hardness on
bounded-degree graphs.
3.1 Introduction
Given an undirected graph G = (V, E), a metric basis of G is a vertex
subset L ⊆ V such that each pair of vertices {u, w} ⊆ V is separated by L,
meaning that there is at least one v ∈ L such that dist(v, u) 6= dist(v, w).
(Recall that dist(v, u) denotes the length of a shortest path between v and u.)
The corresponding Metric Dimension problem has been independently
introduced by Harary and Melter [HM76] and Slater [Sla75]:
Metric Dimension [GJ79](GT61)
Input: An undirected graph G and an integer k ≥ 1.
Question: Is there a metric basis of G which has size at most k?
45
2
1
0
1
2
0
1
2
3
4
2
1/2
1
2/1
0 0/3
3
2
(a)
1
(b)
2/1
3/0
1/2
2/1
(c)
1/0
3/2
0/1
2/3
1/2
(d)
Figure 3.1: Illustration of properties of a metric basis on maximum-degree two
graphs, that is, paths and cycles. In all the graphs depicted above, black-filled
vertices are selected and the labels next to the vertices provide the distances to
the selected vertices. A vertex on a path (see part (a)) forms a metric basis if
and only if it is one of the two endpoints. We will often use the fact that any two
vertices are separated when lying on a shortest path with any selected vertex as an
endpoint. In a cycle (see part (b)) selecting only one vertex is always insufficient
to form a metric basis. Even two selected vertices on a cycle not necessarily form
a metric basis (see part (c)). Indeed, one can show that any two vertices on an
odd-length cycle are a metric basis whereas two vertices on an even-length cycle
are a metric basis if and only if they do not have maximum distance to each other
(see part (d)).
The metric dimension of graphs (the cardinality of a minimum metric
basis) finds applications in various areas including network discovery & verification [Bee+06], metric geometry [HM76], robot navigation, coin weighing
problems, connected joins in graphs, and strategies for the Mastermind game.
We refer to Cáceres et al. [Các+07], Hernando et al. [Her+10], and Bailey
and Cameron [BC11] for a more comprehensive list of applications and a
more complete bibliography. See Figure 3.1 for an introductory-example for
the metric dimension of paths and cycles.
3.1.1 Related Work
From a computational complexity point of view, Metric Dimension is
known to be NP-hard and there is a linear-time algorithm for trees [KRR96].
It has been shown to admit a 2 log n-approximation [KRR96] and to be
inapproximable within o(ln n), unless P = NP [Bee+06]. Additionally, Hauptmann et al. [HSV12] showed that, unless NP ⊆ DTIME(nlog log n ), there is
no (1 − ) ln n approximation for any > 0. Furthermore, they proved APXhardness on bounded-degree graphs. Díaz et al. [Día+12] showed that Metric Dimension remains NP-hard on planar graphs but becomes polynomialtime solvable on outerplanar graphs. In addition, Epstein et al. [ELW12]
showed polynomial-time algorithms as well as NP-hardness results for the
46
vertex-weighted variant of Metric Dimension on several graph classes.1 For
example, they proved that the (unweighted) Metric Dimension problem is
NP-hard on split and bipartite graphs.
3.1.2 Our Contribution
To prove W[2]-hardness for Metric Dimension we provide a reduction from
the W[2]-complete Bipartite Dominating Set problem [DF99, RS08]. It is
a special variant of the NP-complete Dominating Set problem [GJ79](GT2)
where the input graph is restricted to be a bipartite graph.
Bipartite Dominating Set
Input: A bipartite graph G = (V1 ∪ V2 , E) and an integer h.
Question: Does G have a dominating set of size at most h?
More specifically, our polynomial-time reduction maps any instance (G, h)
to an equivalent Metric Dimension instance (G0 , k) with k = h + 4 and G0
being a bipartite graph with maximum degree three. Since Bipartite
Dominating Set is W[2]-hard [RS08], our reduction proves that Metric
Dimension is W[2]-hard with respect to the solution size k even on bipartite
graphs with maximum degree three. Additionally, by a simple reduction to
Bipartite Dominating Set we prove W[2]-completeness. Notably, in both
of the above mentioned reductions that prove W[2]-completeness, the solution
sets in the corresponding problems admit a one-to-one correspondence and
the solution-size parameter increase by at most four. Furthermore, as both reductions are computable in polynomial time, they are indeed linear-parameter
transformations (see Definition 2.3.12). This reveals a surprisingly strong
similarity in terms of the computational complexity of the two problems,
although there is a big difference between the “non-local” nature of Metric
Dimension and the “locally checkable” Bipartite Dominating Set.
The question on the parameterized complexity of Metric Dimension (on
general graphs) with respect to the solution-size parameter was first posed
by Lokshtanov [Lok09a]; also Díaz et al. [Día+12] pointed to this question.
On the one hand, our W[2]-hardness result shows that, unless FPT = W[2],
Metric Dimension is not fixed-parameter tractable. On the other hand,
an algorithm that tests each size-k vertex subset for being a metric basis
runs in O(nk+2 ) time. However, our reduction together with the result that
Bipartite Dominating Set cannot be solved in no(k) time [Che+05, RS08]
implies that Metric Dimension on bipartite graphs with maximum degree
three cannot be solved in no(k) time, unless FPT = W[1]. Thus the trivial
nO(k) -algorithm is (probably) asymptotically optimal.
1 Indeed,
their NP-hardness results all hold for the unweighted variant as well.
47
As Bipartite Dominating Set cannot be approximated within a factor
of o(log n) unless P = NP [AS03, RS08], it also follows that even on bipartite
graphs with maximum degree three there cannot be an o(log n)-factor approximation for Metric Dimension. This strengthens the APX-hardness result
for bounded-degree graphs [HSV12] and it shows that, unless P = NP, the
2 log n-approximation on general graphs [KRR96] is up to constant factors
also optimal on bipartite bounded-degree graphs.
3.1.3 Organization and Further Notation for Paths
Notation for Shortest Paths. In addition to the graph-theoretic notation
introduced in Section 2.1, in this chapter we will write a path (v1 , . . . , vs )
as v1 − v2 − . . . − vs . If there is a unique shortest path between two vertices v and u, then we write just v − u for this path without listing the
intermediate vertices. Furthermore, we extend the distance function dist,
where dist(u, v) is the length of a shortest path from u to v, by setting
dist(v1 , v2 , . . . , vi ) = dist(v1 , v2 ) + dist(v2 , v3 ) + . . . + dist(vi−1 , vi ) for any
vertex subset {v1 , v2 , . . . , vi }.
Organization. In the next section we describe our reduction and prove its
correctness. We proceed by proving W[2]-completeness (Section 3.3) and,
finally, in Section 3.4 we prove the running time as well as the approximation
lower bound.
3.2 Hardness Reduction
In this section we provide a parameterized reduction (even a linear-parameter
transformation, see Definition 2.3.12) from the W[2]-complete Bipartite
Dominating Set problem [RS08] to Metric Dimension on bipartite
graphs with maximum degree three. We start with introducing the main
ideas and concepts, proceed by formally describing the reduction, and then
prove its correctness.
3.2.1 General Ideas and Concepts of the Reduction
Let (G = (V1 ∪ V2 , E), h) be a Bipartite Dominating Set-instance and
let n = |V1 ∪V2 |. We set V = V1 ∪V2 and fix a numbering V1 = {v1 , v2 , . . . , vp }
and V2 = {vp+1 , . . . , vn } such that for all {vi , vj } ∈ E it holds that j ≥ i + 3.
The last is a requirement that helps to simplify the correctness proof and
the existence of such a numbering can be ensured by increasing h by two,
introducing two degree-zero vertices to V2 , and numbering them by vp+1
48
and vp+2 . Clearly, vp+1 and vp+2 have to be contained in any dominating
set for G and as they are degree-zero vertices they do not lower the size of a
minimum dominating set for the rest of the graph.
Remark. Since the distances from a vertex to all vertices in a closed neighborhood of another vertex differ by at most three, log3 ∆ is a lower bound
on the metric dimension of a graph with maximum degree ∆. Avoiding large
degrees was the main obstacle in the reduction below.
We now construct for the Bipartite Dominating Set instance (G, h) an
equivalent instance (G0 , k) of Metric Dimension with k = h + 4. We start
with a high-level description of the graph G0 : It consists of a skeletal structure
in which we embed a vertex-gadget for each vertex of G and an edge-gadget for
each edge of G. Furthermore, there are four particular vertices in the skeletal
structure that are forced to be in any metric basis and these four vertices
separate all but 2n + 1 vertex pairs in each vertex-gadget. Then, choosing
a vertex in a vertex-gadget separates all of the 2n + 1 vertex pairs in its
own gadget plus all the pairs in the vertex-gadgets that are “adjacent” by an
edge-gadget to the chosen vertex-gadget. In this way choosing a metric basis
in the constructed graph G0 corresponds to choosing a dominating set in G.
Throughout the construction, several times we will connect two vertices {u, v} by a so-called y-path, meaning that we insert a path of length y
from u to v. Therein let y be any fixed integer satisfying
1
y > 4n + 4.
4
(3.1)
In all cases we make sure that a y-path is the unique shortest path between its
endpoints and thus a y-path between u and v is denoted by u − v. Intuitively,
y-paths can be viewed as edges of weight y. We use Equation (3.1) several
times in the correctness proof, however, for the intuition it is enough to think
of y being “large” compared to n and, thus, in order to prove that two paths
have different length it is often enough to show that they contain a different
number of y-paths.
To prove that the constructed graph is bipartite we will provide a proper
2-coloring for G0 using colors black and white. Therein the endpoints of
all y-paths are colored white and this uniquely determines the color of all
other vertices on a y-path. To ensure this coloring of y-paths, in addition to
Equation (3.1), we require y to be even. Furthermore, for some details that
will be explained later on, we require that y/2 is also even.2
2 Setting
y = 32n would satisfy all the requirement on y. However, instead of using an
explicit value for y, to improve the understandability of our argumentation at each
point we will explicitly refer to the needed requirement on y.
49
ut`
utr
ut1
ut2
ut3
ut4
utn
Figure 3.2: Schematic illustration of the top-line in the skeletal structure. A bold
line indicates a y-path and a dotted line indicates a path of length more than one.
For each of the corners {ut1 , utn } there is a P3 whose black-colored middle vertex is
adjacent to the corner.
3.2.2 Formal Description of the Reduction
We now describe the construction of G0 in detail. It consists of a skeletal
structure, a vertex-gadget for each vertex in G, and an edge-gadget for each
edge in G connecting the corresponding vertex-gadgets.
Skeletal structure: The skeletal structure contains 2n vertices ut1 , . . . , utn
and ub1 , . . . , ubn where all consecutive vertex pairs {uti , uti+1 } and {ubi , ubi+1 }
are connected by a y-path (see Figure 3.2 for a schematic illustration).
Hence ut1 , . . . , utn and ub1 , . . . , ubn are colored white. We call the first path
ut1 − ut2 − . . . − utn the top-line and the second path ub1 − ub2 − . . . − ubn the
bottom-line. In addition, we call the vertices ut1 , utn , ub1 , ubn corners. For each
corner add a length-three path, that is, a P3 , and make the corner adjacent
to the middle vertex. Additionally, let ut` be any degree-one vertex in the P3
attached to ut1 and correspondingly let utr , ub` , ubr (`: left side; r: right side)
be degree-one vertices in the P3 ’s attached to utn , ub1 , and ubn , respectively.
These four degree-one vertices separate all but 2n + 1 pairs in each vertexgadget. The entire skeletal structure consists of top- and bottom-line plus
the attached P3 ’s.
Vertex-gadget: For each vertex vi ∈ V we add the vertex-gadget giV
to G0 (see Figure 3.3): Construct a cycle of length 4n + 4 and color it
by alternating black and white. We call the vertices cycle vertices. In
addition, two white cycle vertices ati , abi that have distance exactly 2n + 2
are called the anchors of the vertex-gadget. Connect anchor ati by a y-path
to uti and, symmetrically, connect anchor abi by a y-path to ubi . For one of
the two length-(2n + 2) paths (each having n white vertices) between the
anchors denote the white cycle vertices by c1i , . . . , cni . In the following the
neighborhood of all cycle vertices remain unchanged, except for c1i , . . . , cni
which will be used to connect vertex- and edge-gadgets.3
Remark. The vertex pairs in the vertex-gadgets that are not separated by
3 For
the “u-vertices” (e. g. uti , ubi ) in the skeletal structure, the anchor vertices (e. g.
ati , abi ), and the cycle vertices (e. g. c1i , . . . , cn
i ) the subscript always points to the
corresponding vertex in the vertex numbering V = {v1 , . . . , vn }.
50
ut`
ut1
ut2
uti
utn
at1
at2
ati
atn
2n + 1
ub`
utr
c11
c12
c1i
c1n
c21
c22
c2i
c2n
cn1
cn2
cni
cnn
ab1
ab2
abi
abn
ub1
ub2
ubi
ubn
ubr
Figure 3.3: Schematic illustration of vertex-gadgets and their embedding into the
skeletal structure. A bold line indicates a y-path and a dotted line indicates a path
of length more than one.
{ut` , utr , ub` , ubr } are those pairs of cycle vertices in a vertex gadget that have
the same distance to their anchors (e. g. the two black cycle vertices adjacent
to ati ; corresponds to Figure 3.1 part (c)). Hence, there are exactly 2n + 1
vertex pairs in each vertex-gadget not being separated by {ut` , utr , ub` , ubr }.
Note that the number of cycle vertices in a vertex-gadget corresponds to
the right-hand side of Equation (3.1) which allows to “ignore” the traversal
of vertex-gadgets when counting the length of paths in terms of contained
y-paths.
Edge-gadget: For all edges {vi , vj } ∈ E with i < j insert an edgeE
gadget gi,j
into G0 (see Figure 3.4): Add a path of length (j − i + 32 )y
between the two vertices cji and cij . Note that by our requirement that y/2
is even this path is of even length which is necessary because of its white
endpoints. Denote with w`i,j the white vertex on the path having distance y
to cji and denote with wri,j the white vertex on the path having distance y
i,j
to cij . Furthermore, denote with ui,j
` (ur ) the white vertex in the top-line
51
uti
2j
(j − i)y − 2i − 2j
ui,j
`
2i
TL
ati
BL
w`i,j
M
c1j
BR
cji
cni
wri,j
(j − i −
1
2 )y
utj
atj
TR
c1i
cij
cnj
abj
abi
ubi
ui,j
r
(j − i)y
ubj
E
Figure 3.4: A schematic illustration of an edge-gadget gi,j
for the edge {vi , vj }. A
bold line indicates a y-path and a dotted line indicates a path of length more than
one. The concrete length of these “dotted” paths is indicated by the labels next to
the line. The edge-gadget consists of the five parts denoted by BL (bottom left),
BR (bottom right), M (middle), TL (top right), and TR (top right).
that lies between uti and utj and has distance 2j to uti (distance 2i to utj ).
i,j
Then connect w`i,j by a y-path to ui,j
` and also connect wr by a y-path
i,j
0
to ur . This completes the construction of G . By our black/white coloring
the constructed graph is bipartite. Additionally, observe that the maximum
degree in any graph constructed by our reduction is three.
Remark. Note that the construction is “left-right” symmetric, that is, the
distances from {ut` , ub` } are symmetric to those from {utr , ubr }. This does not
not hold from top to bottom since edge-gadgets have y-paths to the top- but
not to the bottom-line.
3.2.3 Correctness of the Reduction
Let (G, h) be an instance of Bipartite Dominating Set and let (G0 , k)
with k = h + 4 be the corresponding instance of Metric Dimension that
is constructed by the reduction described in Section 3.2.2. Clearly, the
reduction is polynomial-time computable and thus it remains to show that G
has a dominating set of size h if and only if G0 has a metric basis of size k.
52
We first give an informal description of the basic ideas.
Outline of the Proof. First, observe that one has to choose at least one
of the two degree-one vertices in the P3 ’s attached to each of the corners of
the top- and bottom-line into any metric basis and a minimum-size metric
basis would never take both. We shall show that {ut` , utr , ub` , ubr } separate
each vertex pair in G0 except those pairs of cycle vertices which have the
same distance to their anchors. Towards this the main observation is that
a shortest path from a vertex in the skeletal structure to a vertex that is
either in a vertex-gadget or also in the skeletal structure would never enter
E
an edge-gadget. For example, traversing an edge-gadget gi,j
by entering it
i,j
3
i
at u` and leaving it at cj gives a path of length (j − i + 2 )y (see Figure 3.4).
However, the path uti − utj − atj − cij that follows the top-line is of length
at most (j − i + 1)y + 2n and, thus, is shorter (recall that 14 y > 4n + 4
by Equation (3.1)). From this the separation of the vertices in the skeletal
structure and most of the vertices in the vertex-gadgets can be deduced.
Because a shortest path starting in one of {ut` , utr , ub` , ubr } has to enter a
vertex-gadget giV always via the anchors {ati , abi }, these vertices cannot
separate pairs of cycle vertices in giV that have the same distance to the
anchors. The fact that {ut` , utr , ub` , ubr } separates each vertex pair consisting
of any two edge-gadget vertices is far from being obvious and proving it
requires to consider several cases (see Lemma 3.2.6 & Lemma 3.2.7). In fact,
the only reason for the additional connection of each edge-gadget to the
top-line (see Figure 3.4, the parts labeled TL and TR) is to separate the
vertices in different edge-gadgets with the four mentioned vertices.
Moreover, we prove that choosing for each cycle vertex in a metric basis
of G0 the vertex that corresponds to the vertex-gadget of this cycle vertex
forms a dominating set in G. Towards this it is crucial that the constant 32
in the definition of edge-gadgets is between one and two: Clearly, taking c1i
into a metric basis separates all cycle pairs in its own gadget. The key point
E
is that it separates also all pairs in gjV if the edge-gadget gi,j
exists: A path
1
V
E
from ci to the cycle vertices of gj via traversing gi,j is of length at most
(j − i + 32 )y + 4n + 4. The “alternative path” via the top-line (via the bottomline is completely symmetric) is by leaving giV via ati , following the top-line
uti − utj , entering gjV via atj and then taking a length at most 2n + 2 path
via the cycle vertices to the desired endpoint. In total, by just counting the
subpath from ati to atj , we infer that this path has length at least (j − i + 2)y.
E
Hence, because 14 y > 4n + 4 (Equation (3.1)), the path traversing gi,j
is
V
shorter. The idea behind is that a path from a cycle vertex in gi to one
in gjV which is not using any edge-gadget has to leave and enter the top- or
bottom-line via the anchors. This costs 2y and since the path uti − utj along
53
the top-line is of length (j − i)y such a path is of length (j − i + 2)y plus
the “short” subpaths through the cycle vertices at start and end. However,
E
the path which uses gi,j
only costs 32 y more than the top- (or bottom-)line
t
t
b
b
path ui − uj (ui − uj ) and thus saves a factor of 12 y from which it is by
Equation (3.1) enough to spend 14 y to argue away the potential difference in
the subpaths through the cycle vertices at start and end. Hence the path
E
via the edge-gadget gi,j
is at least by 14 y shorter.
In addition, choosing c1i into a metric basis only separates pairs in “adjacent”
vertex-gadgets since a shortest path starting in c1i never traverses two edgegadgets. This would cause at least two times the additional cost of 32 y
whereas leaving and entering giV to and from the top-line only costs 2y.
We next give a formal proof of the correctness of the reduction. Towards
this we start with some general observations that will be useful for both
implications of the correctness proof.
General Observations and Additional Notation. We first introduce some
E
additional notation for edge-gadgets. For an edge-gadget gi,j
the four verj i
i,j
i,j
tices {ci , cj , u` , ur } are called entrance-vertices (see Figure 3.4). We
E
partition gi,j
into five parts: The y-path from w`i,j to cji is the BL- (bottom
left) part, the TL- (top left) part is the y-path between w`i,j and ui,j
` , the TR(top right) part is the y-path between wri,j and ui,j
r , and the BR- (bottom
right) part is the y-path between wri,j and cij . Part M (middle) contains the
vertices between w`i,j and wri,j including w`i,j and wri,j .
E
A path enters (leaves) an edge-gadget gi,j
via a vertex v (u) if there are
two consecutive vertices v − w (w − u) on it such that both are contained
E
E
E
in gi,j
and v (u) is an entrance vertex of gi,j
. An edge-gadget gi,j
is traversed
E
by a path P if P contains a subpath consisting only of vertices in gi,j
that
E
E
starts with entering gi,j , contains the M-part, and ends with leaving gi,j
.
E
Observe that there are only four different ways on how to traverse gi,j (see
Figure 3.4) and each is of length (j − i + 32 )y.
We next show that a shortest path that enters an edge-gadget either ends
in it or traverses it. Indeed, as many of the succeeding observations, we will
prove this property even in case of “almost” shortest paths. This allows, after
having a shortest path u − v, to easily argue that for a vertex w “close” to u
the path w − u − v admits the same properties.
Observation 3.2.1. A path that enters and afterwards leaves an edge-gadget
without traversing it is more than 12 y longer than a shortest path with the
same endpoints.
E
Proof. Let P be a path in G0 that starts with entering an edge-gadget gi,j
54
and ends with leaving it but does not traverse it. Since P is simple it is
i,j
j
i,j
i,j
i
equal either to ui,j
` − w` − ci or to ur − wr − cj , implying P being of
i,j
j
t
t
t
t
i
length 2y. However, the paths u` − ui − ai − ci and ui,j
r − uj − aj − cj
are both of length at most y + 4n. Thus P is by at least y − 4n > 21 y
(Equation (3.1)) longer.
Next we prove that any shortest path traverses at most one edge-gadget.
Lemma 3.2.2. If a path is at most 12 y longer than a shortest path with the
same endpoints, then it traverses at most one edge-gadget.
Proof. Assume that there is a path P in G0 that starts with traversing
E
an edge-gadget gi,j
and ends with traversing giE0 ,j 0 and thus has length at
3
0
least (j − i + 2 + j − i0 + 32 )y. In addition, without loss of generality assume
that j 0 ≥ j. We form a path P 0 (possibly neither being a shortest path
nor even simple) with the same endpoints as P that is by more than 12 y
shorter than P .
i,j
Path P 0 starts in the same vertex as P which is one of {cij , cji , ui,j
` , ur }.
j
If P starts in cij (or ci ), then P 0 leaves gjV (giV , resp.) via atj (ati ) to utj
i,j
0
(uti ). From there and also if P starts in one of ui,j
` , ur , path P follows
j
the top-line to utj . Here vertex ci is the worst-case start vertex (yields the
longest path, see Figure 3.4) and thus the “start to utj ”-subpath in P 0 is
of length at most 2j + (j − i + 1)y. To further extend P 0 we make a case
distinction on where P ends. 0 0 0
Case 1: P ends in one of {cji0 , ui` ,j }.
We extend P 0 by the path along the top-line from utj to uti0 . This subpath
is of length (j − i0 )y (by our vertex numbering min{j, j 0 } > max{i, i0 }).
0
From there P 0 either enters giV0 and proceeds till cji0 or it follows the top-line
0 0
till ui` ,j . In both cases the subpath from uti0 to the end is of length at
most y + 2j 0 . In total P 0 is of length at most (j − i + j − i0 + 2)y + 2j + 2j 0
which, by our assumption that j 0 ≥ j and by Equation (3.1), is by more
than 12 y shorter than P .
0 0
0
Case 2: P ends in one of {cij 0 , uir ,j }.
Analogously to Case 1, we extend P 0 by the path along the top-line from utj
0
to utj 0 (length (j 0 − j)y) and from there either via atj 0 to cij 0 or via the
0 0
top-line to uir ,j (length at most y + 2n). In both cases P 0 is of length at
most 4n + (j − i + j 0 − j + 2)y and because of j > i0 (vertex numbering) it
is thus by more than 12 y shorter than P .
We next prove that for each vertex v in G0 , except edge-gadget vertices,
a shortest path from v to any skeletal structure vertex never contains an
edge-gadget vertex.
55
Lemma 3.2.3. In G0 the following properties hold:
(1) For all 1 ≤ i < j ≤ n the path along the top-line (bottom-line) is the
unique shortest path from uti to utj (ubi to ubj ). It has length (j − i)y.
(2) For all 1 ≤ i, j ≤ n the following paths are for all min{i, j} ≤ s ≤
max{i, j} the only shortest paths between uti and ubj : The path
P = uti − uts − ats − c1s − c2s − . . . − cns − abs − ubs − ubj
and the path that results from P by exchanging the subpath c1s − c2s −
. . . − cns by the other path from ats to abs through the cycle vertices of gsV .
Proof. Lemma 3.2.3(1): We prove the claim for i = 1 and j = n on the
top-line. As the vertices for all other choices of i and j also lie on the
top-line, this implies the correctness in all other cases. Lemma 3.2.3(1) can
be analogously proven for the bottom-line.
Assume that there is a shortest path P from ut1 to utn that does not follow
the top-line. We first show that P traverses at least one edge-gadget: Because
the distances on the top- and the bottom-line are completely symmetric,
without entering an edge-gadget a shortest path never starts on the top-line,
enters at some point the bottom-line, and then later on re-enters the top-line.
E
Hence, when leaving the top-line a shortest path enters an edge-gadget gi,j
and, by Observation 3.2.1, it traverses it.
E
Since P is a shortest path and traverses by Lemma 3.2.2 only gi,j
, it
i,j
E
i,j
follows that P enters gi,j via u` and leaves it via ur . This subpath in P
i,j
is of length (j − i + 32 )y. In contradiction to this the path from ui,j
` to ur
along the top-line is of length less than (j − i)y.
Lemma 3.2.3(2): We assume i ≤ j (the other case can be proven completely
analogously). Suppose towards a contradiction that there is a shortest path P 0
from uti to ubj that is different to all paths described in Lemma 3.2.3(2).
By Lemma 3.2.3(1) it holds that if there are two vertices in P 0 that lie on
the top-line (bottom-line), then all vertices on the subpath between them
also lie on the top-line (bottom-line, resp.). Along P 0 from uti to ubj , let utα
be the last vertex on the top-line that corresponds to one of ut1 , ut2 , . . . , utn .
Furthermore, let ubβ be the first vertex on the bottom-line, implying that P 0
contains the y-path abβ − ubβ .
If P 0 traverses an edge-gadget, then the subpath in P 0 from utα to ubβ has
length at least (|β − α| + 32 + 1)y (length of an edge-gadget plus abβ − ubβ ).
In contradiction, the path utα − atα − abα − ubα − ubβ is of length (2 + |β −
α|)y + 2n + 2 and thus is shorter. Hence, P 0 neither traverses and nor
enters (Observation 3.2.1) any edge-gadget and thus α = β. Replacing
56
uti − utα − ubα − ubj in case of α < i by uti − ubi − ubj and in case of α > j by
uti − utj − ubj results in a shorter path. Hence i ≤ α ≤ j, implying that P 0
corresponds to one of the paths described in Lemma 3.2.3(2).
Observation 3.2.1 and Lemmata 3.2.2 & 3.2.3 together with the following
proposition are all what we need to prove the correctness of our reduction
(see Section 3.2.3).
Proposition 3.2.4. The vertices {ut` , utr , ub` , ubr } separate all vertices in G0
except those pairs of cycle vertices from a vertex-gadget that have the same
distance to their anchors.
The next subsection is dedicated to prove Proposition 3.2.4.
Proof of Proposition 3.2.4. The major work in proving Proposition 3.2.4
is to show that the vertices contained in edge-gadgets are separated. Note
that to prove that vertices are separated by {ut` , utr , ub` , ubr } it is sufficient
to separate them by the corners {ut1 , utn , ub1 , ubn }. We use this to simplify
formulas. Additionally, recall that Lemma 3.2.3 provides the distance between
any vertex on the top- or bottom-line to any other vertex either contained
in a vertex-gadget or on the top- or bottom-line. From this the following
E
frequently used distances for any edge-gadget gi,j
can be deduced:
dist(cji , ut1 ) = iy + 2j
dist(cji , utn )
dist(cji , ub1 )
dist(cji , ubn )
t
dist(ui,j
` , u1 )
t
dist(ui,j
` , un )
b
dist(ui,j
` , un )
t
dist(ui,j
r , u1 )
t
dist(ui,j
r , un )
b
dist(ui,j
r , un )
(3.2)
= (n − i + 1)y + 2j
(3.3)
= iy + 2(n − j + 1)
(3.4)
= (n − i + 1)y + 2(n − j + 1)
(3.5)
= (i − 1)y + 2j
(3.6)
= (n − i)y − 2j
(3.7)
= (n − i + 2)y + 2(n − j + 1)
(3.8)
= (j − 1)y − 2i
(3.9)
= (n − j)y + 2i
(3.10)
= (n − j + 2)y + 2(n + i + 1)
(3.11)
For example, we prove Equation (3.8) in detail: The path ut1 − uti − utj − atj −
abj −ubj −ubn is by Lemma 3.2.3(2) a shortest path containing ubn and ui,j
` (it is
i,j
t
t
on the y-path ui − uj , see Figure 3.4). The length of the subpath u` − utj is
(j − i)y − 2j (Figure 3.4), subpath utj − ubj is of length 2y + 2n + 2 (Figure 3.3),
and the subpath from ubj − ubn along the bottom-line is of length (n − j)y.
57
b
Altogether, the subpath ui,j
` − un is of length (n − j + j − i + 2) − 2j + 2n + 2
which can be rearranged to be equal to Equation (3.8).
We first show which entrance vertices of edge-gadgets are used on shortest
paths from one of {ut` , utr , ub` , ubr } to a vertex in the edge-gadget.
Lemma 3.2.5. The following paths are shortest paths and there are no other
shortest paths between any vertex pair contained on one of these paths:
ut1
ub1
utn
ubn
−
−
−
−
ui,j
`
cji
ui,j
r
cij
−
−
−
−
w`i,j
w`i,j
wri,j
wri,j
−
−
−
−
wri,j
wri,j
w`i,j
w`i,j
Proof. Due to the symmetry of the construction we prove Lemma 3.2.5 only
for the paths starting in one of {ut1 , ub1 } (first two paths).
E
Let e ∈ TL ∪ BL ∪ M be a vertex in an edge-gadget gi,j
. We first show
j
t
b
E
that any shortest path from {u1 , u1 } to e enters gi,j via one of {ui,j
` , ci }.
Towards a contradiction, assume that there is a shortest path P from ut1
j
E
or from ub1 to e that enters gi,j
neither via ui,j
` nor via ci , then it has to
contain wri,j . Hence if P starts in ut1 it has length at least
(3.9)
i,j
dist(ut1 , ui,j
r , wr ) = jy − 2i
and if P starts in ub1 it has length at least
(3.4)
dist(ub1 , cij , wri,j ) > (j + 1)y.
In contradiction to this, it holds that
1
i,j j
i,j
i,j
t
i,j (3.6)
dist(ut1 , ui,j
` , w` , ci ) < dist(u1 , u` , w` , wr ) = (j − )y + 2j
2
and also
(3.4)
1
i,j
b j
i,j
dist(ub1 , cji , w`i,j , ui,j
` ) < dist(u1 , ci , w` , wr ) ≤ (j + )y + 2n.
2
Thus P is by at least 41 y longer than a shortest path.
Since
(3.6)
t j
dist(ut1 , ui,j
` ) = dist(u1 , ci ) − y = (i − 1)y + 2j
(3.12)
j
E
and each path from ut1 to e has to enter gi,j
via {ui,j
` , ci } it follows that
i,j
i,j
ut1 − u` − w` − wri,j is a shortest path. Analogously, since
(3.4)
dist(ub1 , cji ) = iy + 2(n − j + 1) < dist(ub1 , ui,j
` ) = (i + 1)y + 2(n + j + 1)
(3.13)
58
j
E
and since each path from ub1 to e has to enter gi,j
via {ui,j
` , ci } it follows
j
i,j
b
i,j
that u1 − ci − w` − wr is a shortest paths.
It remains to argue that for any vertex pair contained on one of these two
paths there exist no other shortest path between them. By Lemma 3.2.3 this
follows for vertex pairs contained on one of the “left subpaths” ut1 − ui,j
` and
j
b
u1 −ci . Moreover, by the construction of the edge-gadgets this also follows for
i,j
j
i,j
vertex pairs contained on one of the “right subpaths” ui,j
` − w` − wr or ci −
i,j
i,j
w` − wr . For vertex pairs where one vertex is contained in the left subpath
and the other in the right subpath it follows from Equations (3.12) & (3.13)
j
and the fact that any path from {ut1 , ub1 } to e is via {ui,j
` , ci } that there is
no other shortest path between them.
Having shown how the entrance vertices of edge-gadgets are used, we next
prove that two vertices contained in the same edge-gadget are separated.
Lemma 3.2.6. The four vertices {ut` , utr , ub` , ubr } separate any two vertices
that are contained in the same edge-gadget.
E
. By
Proof. Let u and v be two vertices contained in an edge-gadget gi,j
i,j
i,j
j
i,j
t
i,j
b
i,j
t
i,j
i,j
Lemma 3.2.5 u1 −u` −w` −wr , u1 −ci −w` −wr , un −ur −wr −w`i,j ,
and ubn − cij − wri,j − w`i,j are shortest paths. Clearly, if u and v are both
contained on one of these paths, then they are separated. Hence, there
remain two cases.
Case 1: u ∈ TL (TR) and v ∈ BL (BR).
Assume u ∈ TL and v ∈ BL. (The other case is completely symmetric.) It
holds that ut1 separates {u, v} because
(3.6)
dist(ut1 , u) = dist(ut1 , ui,j
` , u) ≤ iy + 2j and
i,j
t j
dist(ut1 , v) = min{dist(ut1 , ui,j
` , w` , v), dist(u1 , ci , v)}
(3.2,3.6)
>
iy + 2j.
Case 2: u ∈ TL ∪ BL and v ∈ TR ∪ BR.
It holds that
(3.6)
dist(ut1 , u) ≤ dist(ut1 , ui,j
` , u) ≤ (i − 1)y + 2j + 2y = (i + 1)y + 2j
and, additionally,
t i
t
i,j
dist(ut1 , v) ≥ min{dist(ut1 , ui,j
r ), dist(u1 , cj ), dist(u1 , wr )}
(3.2,3.6,3.9)
=
(3.9)
dist(ut1 , ui,j
r ) = (j − 1)y − 2i.
Since j ≥ i + 3 by our vertex numbering this implies that u and v are
separated by ut1 .
59
To complete the proof of Proposition 3.2.4 it remains to show that vertices
in different edge-gadgets are separated. To this end we make extensive case
distinctions on how the position of the involved edge-gadgets are related to
each other. Moreover, considering just the distance to one of {ut` , utr , ub` , ubr }
will often not be sufficient and thus the general strategy will be to “fix” the
distance to one corner (assume that both vertices have the same distance to it)
and then using this to show that they must be separated by another corner.
Lemma 3.2.7. The four vertices {ut` , utr , ub` , ubr } separate any two vertices
from different edge-gadgets.
E
Proof. Let u be a vertex in the edge-gadget gi,j
(consisting of TL, BL, M,
TR, and BR) and let v be a vertex in the edge-gadget giE0 ,j 0 (consisting of
TL0 , BL0 , M0 , TR0 , and BR0 ). We will assume that the edge-gadgets are
different, implying that i 6= i0 or j 6= j 0 . We prove Lemma 3.2.7 by several
case distinctions. Therein, the following four claims are helpful to simplify
the argumentation.
Claim 1. {u, v} are separated if u ∈ TL ∪ BL and i < i0 . Symmetrically,
{u, v} are separated if u ∈ TR ∪ BR and j > j 0 .
Proof. We prove that at least one of {ut1 , ub1 } separates {u, v} in case of
u ∈ TL ∪ BL and i < i0 . The symmetric claim can be proven analogously
for {utn , ubn }. By Lemma 3.2.5 it follows that all shortest paths from any of
j
E
t
b
{ut1 , ub1 } to u enter gi,j
via {ui,j
` , ci } and among them u1 (u1 ) has minimum
i,j
j
distance to u` (ci ). Hence the following holds for u:
j
i,j
j
b
t
b
dist(ut1 , u, ub1 ) ≤ dist(ut1 , ui,j
` , u, ci , u1 ) ≤ dist(u1 , u` ) + dist(ci , u1 ) + 2y
(3.4,3.6)
=
(i − 1)y + 2j + iy + 2(n − j + 1) + 2y
= (2i + 1)y + 2n + 2.
Now, consider v. Clearly, all shortest paths from ut1 or ub1 to v enter giE0 ,j 0
0
0
0
0
0
0
via one of the entrance vertices {ui` ,j , uir ,j , cij 0 , cji0 }. Among them, ut1 has
0
0
0
minimum distance to ui` ,j and ub1 has minimum distance to cji0 . It follows
that
0
0
0
dist(ut1 , v, ub1 ) > dist(ut1 , ui` ,j ) + dist(ub1 , cji0 )
(3.4,3.6)
=
(i0 − 1)y + 2j 0 + i0 y + 2(n − j 0 + 1)
= (2i0 − 1)y + 2n + 2 ≥ (2i + 1)y + 2n + 2.
Hence, at least one of {ub1 , ut1 } separate {u, v}.
60
Claim 2. {u, v} are separated if i + j 6= i0 + j 0 and either
0
0
0
0
i ,j
3
i,j
1) dist(ui,j
, v, uir ,j ) = (j 0 − i0 + 32 )y,
` , u, ur ) = (j − i + 2 )y and dist(u`
or
0
0
2) dist(cji , u, cij ) = (j − i + 23 )y and dist(cji0 , v, cij 0 ) = (j 0 − i0 + 23 )y.
Proof. In case of 1) it holds that
3
i,j
i,j
i,j
i,j
i,j
dist(ui,j
` , u, ur ) = (j − i + )y = dist(u` , w` , wr , ur )
2
and, hence, that u ∈ TL ∪ M ∪ TR. Thus, it follows that
i,j
t
dist(ut1 , u, utn ) = dist(ut1 , ui,j
` , u, ur , un )
3
((i − 1)y + 2j) + ((n − j)y + 2i) + (j − i + )y
2
1
= 2i + 2j + (n + )y.
2
(3.6,3.10)
=
Symmetrically, dist(ut1 , v, utn ) = 2i0 + 2j 0 + (n + 12 )y. Since i + j 6= i0 + j 0 it
follows that u and v are separated by ut1 and utn .
Now, assume that 2) holds, then, analogously to 1), it holds that
dist(ub1 , u, ubn ) = dist(ub1 , cji , u, cij , ubn )
3
iy + 2(n − j + 1) + (n − j + 1)y + 2(n − i + 1) + (j − i + )y
2
5
= (n + )y + 2(2n − i − j + 2)
2
(3.4,3.5)
=
and, symmetrically, dist(ub1 , v, ubn ) = (n + 52 )y + 2(2n − i0 − j 0 + 2). Since
i + j 6= i0 + j 0 it follows that u and v are separated.
Claim 3. {u, v} are separated if u ∈ M and v ∈ TL0 ∪ TR0 .
Proof. We prove Claim 3 for v ∈ TL0 . The case with v ∈ TR0 follows from
the symmetry of the construction. From Lemma 3.2.5 follows
0
0
(3.6)
dist(ut1 , v) = dist(ut1 , ui` ,j , v) = (i0 − 1)y + 2j 0 + x
0
0
with x = dist(u`i ,j , v) < y. Furthermore,
(3.6)
i,j
i,j
dist(ut1 , u) = dist(ut1 , ui,j
` , w` , u) = iy + 2j + dist(w` , u).
61
Assuming that dist(ut1 , v) = dist(ut1 , u) (otherwise u and v are separated
by ut1 ) we have (i0 − 1)y + 2j 0 + x = iy + 2j + dist(w`i,j , u) and, hence,
dist(w`i,j , u) = (i0 − i − 1)y + 2j 0 − 2j + x.
(3.14)
Thus, i0 ≥ i. From this together with Lemma 3.2.5 it follows that
dist(ub1 , u) = dist(ub1 , cji , w`i,j , u)
(3.4,3.14)
=
iy + 2(n − j + 1) + y + (i0 − i − 1)y + 2j 0 − 2j + x
= i0 y + 2(n − 2j + j 0 + 1) + x.
(3.15)
0
(3.4)
Furthermore, observe that dist(ub1 , v) = dist(ub1 , cji0 , v) = i0 y + 2(n − j 0 +
0
1) + dist(cji0 , v). By Lemma 3.2.5
0
0
0
0
0
0
0
dist(cji0 , v) = min{dist(cji0 , u`i ,j , v), dist(cji0 , w`i ,j , v)}
= min{y + 4j 0 + x, 2y − x}.
(3.4)
Hence, dist(ub1 , v) = i0 y + 2(n − j 0 + 1) + min{y + 4j 0 + x, 2y − x}. Assuming dist(ub1 , u) = dist(ub1 , v) together with Equation (3.15) it follows
that
i0 y + 2(n − 2j + j 0 + 1) + x = i0 y + 2(n − j 0 + 1) + min{y + 4j 0 + x, 2y − x}
and thus 4(j 0 − j) + x = min{y + 4j 0 + x, 2y − x}.
Since the case 4(j 0 −j)+x = y +4j 0 +x yields a contradiction (y = −4j), it
follows that 4(j 0 − j) + x = 2y − x which can be rewritten as x = y − 2j 0 + 2j.
This together with x < y implies j 0 > j and thus i ≤ i0 < j < j 0 . Hence
i + j 6= i0 + j 0 .
To apply Claim 2(1) it remains to show that
0
0
0
0
0
0
0
0
dist(uir ,j , v) = dist(uir ,j , wri ,j , w`i ,j , v),
the rest follows
from
Lemma 3.2.5. Consider the distances with the two
0 0
0 0
0 0
vertices {ui` ,j , w`i ,j } (one of them has to be on a shortest path from uir ,j
to v),
0 0
0 0
0 0
3
1
dist(uir ,j , wri ,j , w`i ,j , v) = (j 0 − i0 + )y − x = (j 0 − i0 + )y + 2j 0 − 2j
2
2
and
0 0
0 0
1
1
dist(uir ,j , ui` ,j , v) = (j 0 −i0 )y−2j 0 −2i0 +x = (j 0 −i0 + )y+ y−4j 0 −2i0 +2j.
2
2
62
0
0
0
0
If a shortest path from uri ,j to v goes via u`i ,j , then 12 y − 4j 0 − 2i0 + 2j ≤
2j 0 − 2j, a contradiction to Equation (3.1). Altogether, the requirements of
Claim 2(1) are fulfilled and thus {u, v} are separated.
Claim 4. {u, v} are separated if u ∈ M and v ∈ BL0 ∪ BR0 .
Proof. We prove Claim 4 for v ∈ BL0 . The case with v ∈ BR0 follows
from the symmetry of the construction. From Lemma 3.2.5 it follows
0
(3.4)
that dist(ub1 , v) = dist(ub1 , cji0 , v) = i0 y + 2(n − j 0 + 1) + x with x =
0
dist(cji0 , v) < y. Furthermore,
(3.4)
dist(ub1 , u) = dist(ub1 , cji , w`i,j , u) = iy + 2(n − j + 1) + y + dist(w`i,j , u)
= (i + 1)y + 2(n − j + 1) + dist(w`i,j , u).
Assuming that dist(ub1 , v) = dist(ub1 , u) (otherwise u and v are separated)
we have i0 y + 2(n − j 0 + 1) + x = (i + 1)y + 2(n − j + 1) + dist(w`i,j , u) and,
hence,
dist(w`i,j , u) = (i0 − i − 1)y + 2(j − j 0 ) + x.
(3.16)
Thus, i0 ≥ i. This together with Lemma 3.2.5 implies
i,j
dist(ut1 , u) = dist(ut1 , ui,j
` , w` , u)
(3.6)
= (i − 1)y + 2j + y + (i0 − i − 1)y + 2(j − j 0 ) + x
= (i0 − 1)y + 2(2j − j 0 ) + x.
In addition, dist(ut1 , v) = dist(ut1 , uti0 , v) = (i0 − 1)y + dist(uti0 , v) and
0
0
0
dist(uti0 , v) = min{dist(uti0 , w`i ,j , v), dist(uti0 , cji0 , v)}
= y + 2j 0 + min{y − x, x}.
Hence, dist(ut1 , v) = i0 y + 2j 0 + min{y − x, x}. Assuming dist(ut1 , u) =
dist(ut1 , v) (otherwise u and v are separated) implies
(i0 − 1)y + 2(2j − j 0 ) + x = i0 y + 2j 0 + min{y − x, x} and
4(j − j 0 ) + x = y + min{y − x, x}.
This gives that either 4(j − j 0 ) + x = 2y − x or 4(j − j 0 ) = y. However,
the second case contradicts Equation (3.1) and, hence, x = y − 2(j − j 0 ).
Since x < y it follows that j > j 0 and, thus, i ≤ i0 < j 0 < j.
63
0
However, since dist(cji0 , v) = x = y − 2(j − j 0 ), it follows that
0
0
0
0
0
0
dist(utn , v) = dist(utn , uri ,j , wri ,j , w`i ,j , v)
1
= (n − j 0 )y + 2i0 + (j 0 − i0 + )y + 2(j − j 0 )
2
1
= (n − i0 + )y + 2(i0 + j − j 0 ).
2
(3.10)
Furthermore,
i,j
i,j
i,j
dist(utn , u) = dist(utn , ui,j
r , wr , w` ) − dist(w` , u)
1
(3.10,3.16)
=
(n − j)y + 2i + (j − i + )y − (i0 − i − 1)y − 2(j − j 0 ) − x
2
1
= (n − i0 + )y + 2i.
2
Hence, dist(utn , u) − dist(utn , v) = 2(i − i0 + j 0 − j). Recall that i ≤ i0
and j 0 < j. Thus, dist(ubn , u) − dist(ubn , v) < 0 and, hence, u and v are
separated by utn .
We now prove Lemma 3.2.7 by a case distinction on how the indices i, i0 , j,
and j 0 are related to each other. Without loss of generality, we assume that
i ≤ i0 . We first prove the case with i < i0 and j = j 0 (Case 1). The case
where i = i0 and j 6= j 0 is omitted because it can be proven completely
analogously. Hence, the remaining cases are i < i0 < j < j 0 (Case 2) and
i < i0 < j 0 < j (Case 3). Note that in all these cases, by Claim 1 we may
assume that u ∈
/ TL ∪ BL.
Case 1 i < i0 < j = j 0 :
If u ∈ M, then Claims 2 to 4 prove that {u, v} are separated. It remains to
consider u ∈ TR ∪ BR.
Subcase 1: u ∈ TR.
Since u ∈ TR by Lemma 3.2.5, Claim 2(1), Equation (3.2), and Equation (3.3)
it follows that
i,j
t
dist(ut1 , u, utn ) = dist(ut1 , ui,j
r , u) + dist(u, ur , un )
= (n − 1)y + 2 · dist(ui,j
r , u).
(3.17)
If v ∈ TR0 , then analogously to Equation (3.17) it follows that
0
dist(ut1 , v, utn ) = (n − 1)y + 2 dist(uri ,j , v).
Hence, assuming dist(ut1 , u, utn ) = dist(ut1 , v, utn ), implies dist(ui,j
r , u) =
0
t
i0 ,j
dist(uir ,j , v). However, dist(ut1 , ui,j
)
=
6
dist(u
,
u
),
implying
that
ut1 separ
r
1
rates {u, v}.
64
It remains to exclude the possibility that v ∈
/ TR0 . Since
(3.10)
dist(utn , u) = dist(utn , ui,j
r , u) ≤ (n − j + 1)y + 2i
(3.18)
and since by Equations (3.3) & (3.7) and our vertex numbering utn has
0
distance at least (n − j + 2)y to both of ui` ,j , cji0 , it follows that a shortest
0
0
path from utn to v enters giE0 ,j via uir ,j or cij . Hence from v ∈
/ TR0 it
follows that
0
0
0
dist(utn , v) ≥ min{dist(utn , uir ,j , wri ,j ), dist(utn , cij )}
(3.10,3.3)
=
(n − j + 1)y + 2i0 .
This together with Equation (3.18) and i < i0 implies that utn separates {u, v}.
Subcase 2: u ∈ BR.
(3.5)
Assume that dist(ubn , u) = dist(ubn , v). Then dist(ubn , u) = dist(ubn , cij , u) <
(n − j + 2)y + 2(n − i + 1). Since i < i0 it follows from Equations (3.5), (3.8)
and (3.11) that v ∈ BR0 . Hence
0
0
dist(ubn , cij , cij , u) = dist(ubn , cij , v) and
0
2(i0 − i) + dist(cij , u) = dist(cij , v).
(3.19)
Denote with x = dist(cij , u).
Subcase 2.1: x ≤ 12 y − 2(i0 + i).
i,j
t
i
Since dist(utn , ui,j
r , wr ) = dist(un , cj )
(3.3,3.10)
=
(n − j + 1) + 2i, from x ≤ 21 y
0
(3.19)
it follows that dist(utn , u) = dist(utn , cij ) + x. Since dist(cij , v) = x +
0
2(i0 − i) ≤ 12 y by the same argument we get dist(utn , v) = dist(utn , cij , v) =
0
0
dist(utn , cij ) + 2(i0 − i) + x, implying that dist(utn , cij ) < dist(utn , cij ).
Subcase 2.2: x > 12 y − 2(i0 + i).
Then
3
t (3.6)
dist(u, ut1 ) = dist(u, wri,j , w`i,j , ui,j
` , u1 ) = (i − 1)y + 2j + (j − i + )y − x
2
1
= (j + )y + 2j − x.
2
Furthermore,
0
3
(3.6)
dist(v, ut1 ) = dist(v, u`i ,j , ut1 ) = (i0 − 1)y + 2j + (j − i0 + )y − x + i − i0
2
1
= (j + )y + 2j − x + i − i0 .
2
65
Since i 6= i0 it follows that u and v are separated by ut1 .
Case 2 i < i0 < j < j 0 :
If u ∈ M, then Claim 2 (v ∈ M0 ), Claim 3 (v ∈ TL0 ∪ TR0 ), and Claim 4
(v ∈ BL0 ∪ BR0 ) prove that {u, v} are separated. It remains to consider
u ∈ TR ∪ BR. By Claim 1 (for v ∈ TR0 ∪ BR0 and j 0 > j), Claim 3 (u ∈ TR
and v ∈ M 0 ), and Claim 4 (u ∈ BR and v ∈ M0 ) it remains to consider the
case where v ∈ TL0 ∪ BL0 .
Subcase 1: u ∈ TR.
(3.5)
It follows that dist(ubn , u) > dist(ubn , abj , cij ) + y = (n − j + 2)y + 2(n − i + 1)
and if v ∈ BL0 , then
(3.5)
0
dist(ubn , v) ≤ dist(ubn , abj0 , cij 0 ) + y = (n − j 0 + 2)y + 2(n − i0 + 1).
Since j 0 > j it follows that ubn separates {u, v} if v ∈ BL0 .
It remains to consider v ∈ TL0 . By Lemma 3.2.5
(3.10)
dist(utn , u) = dist(utn , ui,j
r , u) = (n − j)y + 2i + x
0
0
(3.7)
i ,j
t
t
where x = dist(u, ui,j
, v) =
r ) < y. Furthermore, dist(un , v) = dist(un , u`
0 0
i
,j
(n − i0 )y − 2j 0 + dist(u` , v). Hence assuming dist(utn , u) = dist(utn , v) we
get j = i0 + 1 and
0
0
(n − j)y + 2i + x = (n − j + 1)y − 2j 0 + dist(u`i ,j , v)
0
0
dist(ui` ,j , v) = x + 2i + 2j 0 − y.
Since x < y it follows that
0
0
(3.6)
dist(ut1 , v) = dist(ut1 , ui` ,j , v) = (i0 − 1)y + 2j 0 + x + 2i + 2j 0 − y
(3.1)
= (j − 3)y + 4j 0 + 2i + x < (j − 1)y − 2i
(3.9)
t
= dist(ut1 , ui,j
r ) < dist(u1 , u).
Thus, u and v are separated by ut1 .
Subcase 2: u ∈ BR.
(3.5)
From u ∈ BR it follows that dist(ubn , u) = dist(ubn , cij , u) = (n − j + 1)y +
2(n − i + 1) + x where x = dist(u, cij ) < y. By an argumentation analogously
to the first part of Subcase 1, it follows that it remains to consider the case
where v ∈ BL0 . In addition, it holds that
0
(3.5)
0
dist(ubn , v) = dist(ubn , cji0 , v) = (n − i0 + 1)y + 2(n − j 0 + 1) + dist(cji0 , v).
66
Assuming dist(ubn , u) = dist(ubn , v), we have j = i0 + 1. Hence,
0
(n − j + 1)y + 2(n − i + 1) + x = (n − j + 2)y + 2(n − j 0 + 1) + dist(cji0 , v)
0
dist(cji0 , v) = x − 2i + 2j 0 − y.
0
Since x < y it follows that dist(cji0 , v) < j 0 − i. Hence,
0
(3.4)
dist(ub1 , v) = dist(ub1 , cji0 , v) = i0 y + 2(n − j 0 + 1) + x − 2i + 2j 0 − y
(3.1)
= (j − 2)y + 2(n − i + 1) + x < jy + 2(n − i + 1)
(3.4)
= dist(ub1 , cij ) < dist(ub1 , u).
Thus, u and v are separated by ub1 .
Case 3 i < i0 < j 0 < j:
If u ∈ TR ∪ BR, then since j 0 < j it follows from Claim 1 that {u, v} are separated. It thus remains the case where u ∈ M. If v ∈ TL0 ∪ TR0 ∪ BL0 ∪ BR0 ,
then Claims 3 & 4 prove that {u, v} are separated. Thus let v ∈ M0 and
i + j = i0 + j 0 (otherwise they would be separated due to Claim 1).
0
0
(3.6)
From dist(u, ut1 ) = dist(v, ut1 ) it follows dist(ut1 , w`i ,j ) − dist(ut1 , w`i,j ) =
0 0
(i0 − i)y + 2j 0 − 2j, implying dist(w`i ,j , v) = dist(w`i,j , u) − 2j 0 + 2j − (i0 − i)y.
This leads to
0
0
0
dist(ub1 , v) = dist(ub1 , cji0 , w`i ,j , v)
0
= dist(ub1 , cji0 ) + y + dist(w`i,j , u) − 2j 0 + 2j − (i0 − i)y
(3.4) 0
= i y + 2(n − j 0 + 1) + y + dist(w`i,j , u) − 2j 0 + 2j − (i0 − i)y
= (i + 1)y + 2(n + j − 2j 0 + 1) + dist(w`i,j , u).
By Equation (3.4) it holds that dist(ub1 , u) = dist(ub1 , cji ) + y + dist(w`i,j , u) =
iy + 2(n − j + 1) + y + dist(w`i,j , u). Thus,
dist(ub1 , u) − dist(ub1 , v) = iy + 2(n − j + 1) + y + dist(w`i,j , u)
− ((i + 1)y + 2(n + j − 2j 0 + 1) + dist(w`i,j , u))
= 4j 0 − 4j.
Since j 6= j 0 it follows that u and v are separated by ub1 .
Having proved Lemmata 3.2.6 & 3.2.7, we now have all the ingredients to
prove Proposition 3.2.4 (p. 57). It claims that {ut` , utr , ub` , ubr } separate all
vertices in the constructed graph G0 except those pairs of cycle vertices from
a vertex-gadget that have the same distance to their anchors.
67
Proof of Proposition 3.2.4: We will show that for each pair of vertices in G0 ,
except those pairs of cycle vertices from a vertex-gadget that have the same
distance to their anchors, there is a vertex in {ut` , utr , ub` , ubr } that separates it.
We have three groups of vertices in G0 namely vertex-gadget-vertices, edgegadget vertices, and vertices in the skeletal structure. Next, we shall show
that each of them is separated from all others.
Skeletal vertices: We prove that all vertices on the top- and bottom-line
are separated from all others. First, for each vertex pair from the skeletal
structure by Lemma 3.2.3 there is a shortest path between two vertices from
{ut` , utr , ub` , ubr } that contains both vertices, implying that they are separated.
Next, consider a vertex pair {u, v} where u is contained in the skeletal
structure and v is contained in a vertex-gadget gsV . More specifically, let u be
on the top-line on the y-path uti − uti+1 (the proof is completely analogous
for the bottom-line). If i < s, then by Lemma 3.2.3 the following path is a
shortest path ut` − uti+1 − uts − ats − abs − ubs − ubs+1 − ubr (ats − abs is through v
if v is cycle vertex, otherwise take any of the both paths through the cycle
vertices) which contains u and v. Symmetrically, if i ≥ s, then the following
is a shortest path ub` − ubs − abs − ats − uts − uti − utr (again abs − ats is via v
if v is a cycle vertex). In both cases the vertex pair lies on a shortest path
starting in {ut` , utr , ub` , ubr } and thus is separated.
Finally, by Lemma 3.2.3 for each vertex u in the skeletal structure there is
a path from ut` to ubr that contains u and there is no shortest path from ut`
to ubr containing any edge-gadget vertex, implying that u is separated by ut`
or ubr from all edge-gadget vertices.
Vertex-gadget: By the argument above, vertex-gadget vertices are separated from vertices in the skeletal structure. Furthermore, for each vertexgadget vertex v there is a shortest path from ut` to ubr via v and no shortest
path between them contains any edge-gadget vertex (Lemma 3.2.3), implying
that ut` or ubr separate v from all edge-gadget vertices.
It remains to prove that any vertex-gadget vertex v is separated from any
other vertex-gadget vertex v 0 except in the case that they correspond to
a pair of cycle vertices from a vertex-gadget that have the same distance
to their anchors. Consider first the subcase where v and v 0 are contained
in the same vertex-gadget giv . Then, by Lemma 3.2.3(2) the following is a
shortest path: ut` − uti − ati − c1i − c2i − . . . − cni − abi − ubi − ubr and the subpath
c1i − c2i − . . . − cni can be exchanged by the other path from ati to abi through
the cycle vertices of giV . This implies that v and v 0 are separated.
Consider the subcase where v and v 0 are in different vertex-gadgets, say
v ∈ giV and v 0 ∈ gjV with i < j. With x = dist(uti , v) and the assumption
68
dist(ut1 , v) = dist(ut1 , v 0 ) it follows that
0 = dist(ut1 , v) − dist(ut1 , v 0 ) = dist(ut1 , uti ) + x − dist(ut1 , utj ) − dist(utj , v 0 )
= (i − 1)y + x − (j − 1)y − dist(utj , v 0 ),
implying that dist(utj , v 0 ) = (i − j)y + x. From dist(utn , v) = dist(utn , v 0 ) it
follows that
0 = dist(utn , v) − dist(utn , v 0 ) = (n − i)y + x − ((n − j)y + dist(utj , v 0 ))
= (j − i)y + x − (i − j)y − x = (2j − 2i)y
and thus a contradiction since i < j.
Edge-gadget: Because of the above considerations it only remains to prove
that edge-gadget vertices are separated from other edge-gadget vertices. This
is done by Lemmata 3.2.6 & 3.2.7.
Correctness. Based on Observation 3.2.1, Lemmata 3.2.2 & 3.2.3, and Proposition 3.2.4 we next prove the correctness of our reduction (see Section 3.2.1).
For the sake of readability the proof is separated into two implications.
Proposition 3.2.8. If (G, h) is a yes-instance of Bipartite Dominating
Set, then (G0 , k) is a yes-instance of Metric Dimension.
Proof. For a yes-instance (G = (V, E), h) with V = {v1 , . . . , vn } of Bipartite Dominating Set denote by K ⊆ V a dominating set of size at most h.
We prove that the corresponding Metric Dimension instance (G0 , k) with
k = h + 4 is also a yes-instance. More specifically, we prove that the vertex
subset L of G0 that contains {ut` , utr , ub` , ubr } and for each vertex vi ∈ K the
vertex c1i is a metric basis.
By Proposition 3.2.4 the vertices {ut` , utr , ub` , ubr } ⊆ L separate all pairs of
vertices in G0 except pairs of cycle vertices having the same distance to their
anchors. Clearly, each such pair {v, w} in a vertex-gadget giV is separated
if c1i ∈ L. Thus, consider the case where c1i ∈
/ L. As K is a dominating
set there is a vertex c1α ∈ L such that {vi , vα } ∈ E, implying that there
E
E
is an edge-gadget gi,α
(or gα,i
). Next, we prove that c1α separates the pair
{v, w}. This is done by proving that P l = c1α − ciα − wri,α − w`i,α − cα
i − v (if
α < i then interchange w`i,α and wri,α ) is a shortest path and all other paths
between c1α and v are more than 14 y longer. Having proved this it follows
that P l extended by a shortest path within giV is also a shortest path for w.
Thus {v, w} is separated by c1α .
The length of P l is at most 4n + 4 + (|α − i| + 32 )y. By Lemma 3.2.2 each
path that is at most 12 y longer than a shortest path from c1α to v traverses at
69
most one edge-gadget and each path that traverses an edge-gadget different
E
from gi,α
is at least by 2y longer than P l (it has to leave gαV to the top- or
bottom line and then it has to re-enter some vertex-gadget). Thus it remains
to consider the paths from c1α to v that do not traverse any edge-gadget.
There are only two of them: c1α − atα − utα − uti − ati − v (via top-line) and
c1α − abα − ubα − ubi − abi − v (via bottom-line). Both are of length more than
(|α − i)| + 2)y and, thus, are at least 14 y longer than P l (Equation (3.1)).
Thus P l is a shortest path, implying that c1α separates {v, w}.
Proposition 3.2.9. If (G0 , k) is a yes-instance of Metric Dimension,
then (G, h) is a yes-instance of Bipartite Dominating Set.
Proof. Let (G0 , k) be a yes-instance of Metric Dimension where G0 is
constructed from the Bipartite Dominating Set instance (G = (V, E), h)
with k = h + 4 and V = {v1 , . . . , vn }. Furthermore, let L be a metric basis
of G0 of size at most k. As already argued, L contains at least one degree-one
vertex from each of the P3 ’s attached to of each of the corners {ut1 , utn , ub1 , ubn }
(otherwise the two degree-one vertices in a P3 are not separated). Then
Proposition 3.2.4 proves that these degree-one neighbors separate all vertices
in G0 except pairs of cycle vertices having the same distance to their anchors.
We now form a vertex subset K ⊆ V and prove that it is a dominating set
of size at most h: For each vertex v ∈ L in a vertex-gadget giV add vi ∈ V
E
to K. Additionally, for each vertex v ∈ L contained in an edge-gadget gi,j
E
add vi to K if v is contained on the TL- or BL-part of gi,j and add vj to K
in all other cases. Clearly, |K| ≤ |L| − 4.
We next prove that K is a dominating set for G. Suppose towards a
contradiction that there is a vertex vi ∈ V that is not dominated by K. By
definition of K none of the vertices in giV is contained in the metric basis L.
However, there is one vertex u ∈ L that separates the pairs of cycle vertices
in giV having to same distance to ati , abi . Let {v, w} be one of these pairs
and without loss of generality let w be the vertex contained on the path
ati − c1i − . . . − cni − abn . (Hence v is on the other path from ati to abi .) Denote
by P v the set of all shortest paths from u to v and by P w the set of all
shortest paths from u to w. By the construction of the vertex-gadget it
follows that each path in P v either contains ati or abi . Since v and w have the
same distance to ati and abi it follows that all paths in P w neither contain ati
nor abi , since otherwise v and w would not be separated by u. Hence, each
E
path in P w enters giV via an entrance vertex cji of an edge-gadget gi,j
. If u is
contained either directly in one of these edge-gadgets or it is contained in gjV ,
then by the construction of K this implies that either vi or vj is contained
in K. This yields a contradiction since {vi , vj } ∈ E and thus vi is dominated.
In the remaining case, towards a contradiction, consider a shortest path
70
E
in P ∈ P w entering giV via cji but u is neither contained in gi,j
nor in gjV .
E
Clearly, by Observation 3.2.1 it follows that P traverses gi,j . Thus, P eni,j
E
i
ters gi,j
via ui,j
r (u` if i > j) or cj . However, by Lemma 3.2.3 the (unique)
i,j
j
i,j
shortest path from ur (u` ) to ci contains ati , implying a contradiction in the
E
first case. Hence, we can assume that P enters gi,j
via cij . By Lemma 3.2.2
E
it traverses only gi,j
, implying that it enters gjV either via an anchor or via
α
some cj . If P enters gjV via the anchor atj (abj ) this implies that P contains utj
(ubj ). However, by Lemma 3.2.3 the (unique) shortest path from utj (ubj ) to w
contains ati (abi ), yielding a contradiction. In the remaining case the path
E
from u to w enters gjV via cα
j and since it traverses only gi,j , this implies
E
that u is contained in gj,α . In addition, by the construction of K it follows
E
that u has distance greater than y to cα
j (it is not in TL ∪ BL of gj,α ) and,
hence, P contains w`j,α (if j < α) or wrj,α in case of j > α. The subpath
from w`j,α or from wrj,α to w is of length at least (1 + |j − i| + 32 )y. However,
the vertex in {w`i,j , wri,j } from P that is closest to cα
j has distance at most
y + 2n to utj and dist(utj , c1i ) = (|j − i| + 1)y + 2n + 1, implying that P is
not a shortest path.
Proposition 3.2.8 together with Proposition 3.2.9 imply that our reduction
given in Section 3.2.1 is correct. In the remaining part we discuss the
consequences that are implied by it.
3.3 W[2]-Completeness
In the previous section we proved the correctness of our polynomial-time
reduction which maps an instance (G, h) of Bipartite Dominating Set
into an instance (G0 , k) of Metric Dimension with k = h + 4. Since
Bipartite Dominating Set is W[2]-hard with respect to h [RS08], this
implies that Metric Dimension is W[2]-hard with respect to k on bipartite
graphs with maximum degree three. Note that this classification is tight in
the sense that Metric Dimension is (trivially) polynomial-time solvable
on graphs with maximum degree two (Figure 3.1). We prove in this section
that Metric Dimension is indeed W[2]-complete.
Theorem 3.3.1. Metric Dimension on bipartite graphs with bounded degree three is W[2]-complete with respect to the parameter size of a metric basis.
Proof. The W[2]-hardness follows from the parameterized reduction provided
in Section 3.2.2. Hence, it remains to show containment in W[2]. This is
done by giving a parameterized reduction from Metric Dimension to
71
Bipartite Dominating Set (the reduction is similar to that of Raman and
Saurabh [RS08] from Dominating Set to Bipartite Dominating Set):
Let (G, k) be an instance of Metric Dimension. We form a bipartite
graph G0 = (P ∪ D, E 0 ) such that (G0 , k + 1) is a Bipartite Dominating
Set instance which is equivalent to (G, k). Construct G0 by inserting for each
vertex pair {u, w} ⊆ V (G) a vertex vu,w to P and for each vertex u ∈ V (G)
insert a corresponding vertex to D. Furthermore, add an edge from a vertex t
in D to a vertex vu,w ∈ P if the vertex in V (G) which corresponds to t
separates in G the vertex pair {u, w}. In this way choosing a vertex from D
into a dominating set of G0 corresponds to choosing this vertex in G into a
metric basis and vice versa.
We further add a vertex zD to D and a vertex zP to P such that zD is a
degree-one vertex with neighbor zP and zP is adjacent to all vertices in D.
Because of its degree-one neighbor zD one may assume that zP is contained
in any dominating set of G0 , implying that all vertices in D are dominated
by this vertex. Hence, besides zP , each dominating set of G0 can be assumed
to be a subset of D dominating the vertices in P and thus corresponds to a
metric basis in G.
3.4 Running Time and Approximation Lower
Bounds
We next show a running time as well as an approximation lower bound for
Metric Dimension.
Chen et al. [Che+05] proved that Dominating Set (given an n-vertex
graph, decide whether it has a size-h dominating set) cannot be solved
in no(h) time, unless FPT = W[1]. By the details of the reduction from
Dominating Set to Bipartite Dominating Set provided by Raman
and Saurabh [RS08] (there is a one-to-one correspondence between the
solution sets) this result also holds for Bipartite Dominating Set. This
implies, together with the observation that the parameter k in our reduction
(see Section 3.2.1) is linearly upper-bounded by the parameter h from the
Bipartite Dominating Set instance where we reduce from, the same
running time lower bound for Metric Dimension.
Theorem 3.4.1. Unless FPT = W[1], Metric Dimension cannot be solved
in no(k) time, even on bipartite graphs with maximum degree three.
Note that the lower bound provided by Theorem 3.4.1 is asymptotically
tight in the sense that a trivial brute-force algorithm that tests each size-k
vertex subset whether it is a metric basis achieves a running time of O(nk+2 ).
72
Patrascu and Williams [PW10] improved the result of Chen et al. [Che+05]
by showing that Dominating Set cannot be solved in O(nk− ) time for
every > 0, unless the SETH (Hypothesis 2.3.17) fails. Since our reduction
has a blow up in the number of vertices which can be upper-bounded by O(n5 )
and only a constant additive blow up in the solution size parameter, it
k
follows that Metric Dimension cannot be solved in O(n 6 ) time, unless
the SETH fails.
Observe that the proof of Proposition 3.2.9 also provides a one-to-one
correspondence between a metric basis and a dominating set in the instance
where we reduce from. In addition, our reduction can be computed in
polynomial time. The reduction from Dominating Set to Bipartite
Dominating Set [RS08] also has these two properties. Thus, the result that
Dominating Set cannot be approximated within o(log n) within polynomialtime, unless P = NP [AS03], transfers to Metric Dimension.
Theorem 3.4.2. Unless P = NP, Metric Dimension on bipartite graphs
with maximum degree three cannot be approximated in polynomial-time within
a factor of o(log n).
3.5 Conclusion and Open Questions
We showed that Metric Dimension is W[2]-complete even on bipartite
graphs with maximum degree three. We performed a first step towards a
systematic study of the parameterized complexity of Metric Dimension.
The most interesting questions that arise are whether Metric Dimension is
fixed-parameter tractable on planar graphs and with respect to the treewidth
of the input graph.
Towards a systematic study of structural parameterizations, by simple
observations on vertices with the same neighborhood (at most one of them
can be absent in a metric basis), it is straightforward to prove the fixedparameter tractability of Metric Dimension with respect to the size of
a vertex cover. The size of a metric basis can be arbitrarily large even in
trees (for example, in a star one has to take all but two vertices). However,
Epstein et al. [ELW12] have shown that there is a data reduction rule such
that after its exhaustive application for any input graph it holds that the
size of a minimum-size metric basis is upper-bounded in a linear function of
the size of a feedback edge set. Recalling the second approach to identify
tractable cases of NP-hard problem (Chapter 1), this motivates a systematic
study of stronger parameterizations (Definition 2.3.14), for example the size
of a feedback vertex set. See Figure 3.5 for an overview of the relation
between these parameters.
73
FPT
vertex cover
feedback edge set
feedback
vertex set
metric dimension
maximum degree
W[2]-hard
distance to
bipartite
treewidth
NP-hard with constant
parameter values
degeneracy
Figure 3.5: Overview of some structural parameterizations of Metric Dimension on graphs that are reduced with respect to a data reduction rule due to
Epstein et al. [ELW12]. An edge from a higher-drawn parameter α down to a parameter β indicates that β is a stronger parameterizations than α (Definition 2.3.14,
the diagram is an image detail of Figure 2.1). Because of Epstein et al.’s [ELW12]
data reduction rule it follows that the metric dimension is upper-bounded by a
linear function in the size of a feedback edge set. For two vertices with the same
neighborhood it holds that one of them has to be contained in each metric basis.
Based on this it is straightforward to show that Metric Dimension is solvable in
k
O(22 +k · nO(1) ) time with k denoting the size of a vertex cover. Our reduction
shows W[2]-hardness with respect to the size of a metric basis. Moreover, since
the constructed graph is bipartite and has maximum degree at most three, the
NP-hardness results for the parameters distance to bipartite, max. degree, and
degeneracy follow. The parameterized complexity with respect to the parameters
feedback vertex/edge set and treewidth is open.
Finally, it is open whether the 2o(n) running time lower bound for Dominating Set (based on the ETH, see Hypothesis 2.3.15) [LMS11] can be
transfered to Metric Dimension.
74
4 Parameterizing 2-Club by
Structural Graph Parameters
In this chapter we examine the computational complexity of the NP-complete
2-Club problem. The problem is, given a graph G = (V, E) and ` ∈ N, to
decide whether there is a vertex set S ⊆ V of size at least ` such that G[S]
has diameter at most two. 2-Club has applications in the analysis of social
and biological networks.
We provide a systematic classification of the parameterized complexity of
2-Club with respect to a hierarchy of prominent structural graph parameters
(similarly to Figure 2.1 on p. 43). That is, we systematically explore the
structural parameter space of the input graph and examine the influence of
the contained parameters on the tractability of 2-Club.
Going from bottom to top in this parameter hierarchy, we start in Section 4.2 to prove several intractability results. For example, we prove that
2-Club is NP-hard on graphs that become bipartite by deleting one vertex
and on graphs with degeneracy five. Furthermore, it is W[1]-hard with
respect to the parameter h-index. This parameter is motivated by real-world
instances and the fact that 2-Club is fixed-parameter tractable with respect
to the weaker parameter maximum degree.
Having identified NP-hard cases towards drawing the border of intractability, we provide several tractability results (see Section 4.3): We prove that
2-Club is fixed-parameter tractable with respect to the parameter distance to cographs which measures how many vertices have to be deleted
to transform the input graph into a cograph. We also provide a direct
combinatorial algorithm for the parameter treewidth. So far this algorithm is,
up to some constant factors in the exponent, the best performing algorithm
even for weaker parameters such as vertex cover.
We also show for which parameters in the hierarchy 2-Club is admitting a
polynomial kernel (Section 4.4). We prove their existence for the parameters
cluster editing set and feedback edge set. However, for a large part of the
parameter hierarchy we prove the non-existence of polynomial kernels unless
NP ⊆ coNP/poly.
Notably, the parameter distance k 0 to a 2-club which measures how many
vertices have to be deleted to transform the input graph into a 2-club,
although not admitting a polynomial kernel, admits a simple search tree
75
0
algorithm running in O(2k · nm) time [Sch+12]. Under the condition that
the SETH (Hypothesis 2.3.17) does not fail, we exclude the existence of an
0
algorithm running in O((2−)k ·(nm)O(1) ) time for all > 0 (see Section 4.5).
Hence we prove that the known algorithm is essentially tight. Finally,
somewhat in contrast to this lower bound we demonstrate that the insights
obtained from (Turing) kernelization algorithms can be exploited to design
0
efficient data reduction rules that, together with the O(2k · nm)-time search
tree algorithm, develop into an efficient implementation of an exact algorithm
for 2-Club that outperforms previous implementations (see Section 4.6).
4.1 Introduction
The identification of cohesive subnetworks is an important task in the
analysis of social and biological networks since these subnetworks are likely
to represent communities or functional subnetworks within the large network.
The natural cohesiveness requirement is to demand that the subnetwork is a
complete graph, a clique. However, this requirement is often too restrictive
and thus relaxed definitions of cohesive graphs such as s-cliques [Alb73],
s-plexes [SF78], and s-clubs [Mok79] have been proposed. In this work,
we study the problem of finding large s-clubs within the input network.
An s-club is a vertex set that induces a subgraph of diameter at most s.
Thus, s-clubs are distance-based relaxations of cliques, which are exactly the
graphs of diameter one. For constant s ≥ 1, the problem of finding s-clubs
is defined as follows.
s-Club
Input: An undirected graph G = (V, E) and ` ∈ N.
Question: Is there a vertex set S ⊆ V of size at least ` such that G[S] has
diameter at most s?
Observe that 1-Club is equivalent to Clique. In this work, we study
the computational complexity of 2-Club, that is, the special case s = 2
(see Figure 4.1). This is motivated by the following two considerations.
First, 2-Club is an important special case occurring in many real-world
applications: For biological networks, 2-clubs and 3-clubs have been identified
as the most relevant diameter-based relaxations of cliques [Pas08]. Further,
Balasundaram et al. [BBT05] proposed to compute 2-clubs and 3-clubs for
analyzing protein interaction networks. 2-Club also has applications in the
analysis of social networks [ML06]. Consequently, extensive experimental
studies concentrate on finding 2- and 3-clubs [AC12, BBT05, BLP02, CA11,
Cha+13, PB12]. The second reason for studying 2-Club is that it is the
most basic variant of s-Club which is different from Clique. For example,
76
C5
star
diamond
Figure 4.1: An illustration of prominent graphs which are 2-clubs. As the “star”graph shows that a vertex with all its neighbors is a 2-club, in non-trivial instances
it holds that ` > ∆ + 1 for the maximum degree ∆. Additionally, all three graphs
illustrate that in difference to cliques, being a 2-club is non-hereditary. Furthermore,
since a C5 , a cycle with five vertices, is the largest cycle which is a 2-club it follows
that if the length of a shortest cycle in a graph (the so-called girth) is at least six,
then the largest 2-club is a vertex along with its neighborhood.
being a clique is a hereditary graph property, that is, it is closed under
vertex deletion. In contrast, being a 2-club is not hereditary, since deleting
vertices can increase the diameter of a graph. Hence, it is interesting to spot
differences in the computational complexity of the two problems.
We aim to describe how structural properties of the input graph determine
the computational complexity of 2-Club (see Section 2.3.5 for an introduction to structural parameterization). We want to determine sharp boundaries
between tractable and intractable special cases of 2-Club, and whether some
graph properties, especially those motivated by the structure of social and
biological networks, can be exploited algorithmically. By arranging the parameters in a hierarchy (ranging weaker parameterizations down to stronger
parameters, see Definition 2.3.14) we draw a border line between tractability
and intractability to obtain a systematic view (this hierarchy forms the
structural parameter space). A similar approach was followed for other
hard graph problems such as Odd Cycle Transversal [JK12], Target
Set Selection [Cho+13], and for the computation of the pathwidth of a
graph [BJK12]. The hierarchy of structural graph parameters is inspired by
work of Jansen [Jan11],[Jan13]. An extended version was collected by Sorge
and Weller [SW13].
4.1.1 Related Work
For all s ≥ 1, s-Club is NP-complete on graphs of diameter s + 1 [BBT05];
2-Club is NP-complete even on split graphs and, thus, also on chordal
77
graphs [BBT05].1 In contrast, 2-Club is solvable in polynomial time on bipartite graphs, on trees, and on interval graphs [Sch09]. Golovach et al. [Gol+13]
further explored the complexity of s-Club in special graph classes. For example they proved polynomial-time solvability of s-Club on choral bipartite,
strongly chordal and distance hereditary graphs. Additionally, it is proven
that on a superclass of these graph classes, called weakly chordal graphs,
s-Club is polynomial-time solvable for odd s and NP-complete for even s.
s-Club is well-understood from the viewpoint of approximation algorithms [AMS10]: It is NP-hard to approximate s-Club within a factor of
1
n 2 − for any > 0. On the positive side, it has been shown that
a largest set
consisting of a vertex together with all vertices within distance 2s is a factor
1
2
n 2 approximation for even s ≥ 2 and a factor n 3 approximation for odd s ≥ 3.
Several heuristics [BLP00, CA11, Cha+13], integer linear programming formulations [AC12, BBT05, BLP02], parameterized algorithms [Sch+12], and
branch-and-bound algorithms [BLP02] have been proposed and experimentally evaluated [AC12, BBT05, BLP02, CA11, Cha+13, PB12].
From the viewpoint of parameterized algorithmics, 1-Club is equivalent
to Clique and thus W[1]-hard with respect to the solution size ` [DF99].
In contrast, for all s ≥ 2, s-Club is fixed-parameter tractable with respect
to ` [Sch09, Sch+12] and also with respect to the parameter treewidth [Sch09].
Moreover, s-Club does not admit a polynomial kernel with respect to `
(unless NP ⊆ coNP/poly) [Sch+12]. Taking for each vertex the vertex itself
together with all other vertices that are in distance at most s yields a Turing
kernel (Definition 2.3.8) with at most `2 vertices for even s and at most `3
vertices for odd s [Sch+12]. The main observation behind the parameterized
algorithm for ` is that any closed neighborhood N [v] of a vertex v is an
s-club for s ≥ 2. Hence, the maximum degree ∆ in non-trivial instances is
less than ` − 1. It also holds, however, that ` ≤ ∆s + 1 in yes-instances. Thus,
for constant s, fixed-parameter tractability with respect to ` also implies
fixed-parameter tractability with respect to the maximum degree.
In addition, a search tree-based algorithm that branches into the two
possibilities to delete one of two vertices with distance more than s achieves
a running time of O(2n−` · nm) for the dual parameter n − ` which measures
the distance to an s-club [Sch+12].2 Interestingly, Chang et al. [Cha+13]
proved that with respect to the number of vertices n the same search tree
algorithm runs in O(αn · nm) time with the golden ratio α < 1.62.
1 An
2
78
NP-hardness reduction given by Balasundaram et al. [BBT05, Theorem 1] can be easily modified such that the 2-Club instance is a split graph (make vertex set E a clique).
Schäfer et al. [Sch+12] considered finding an s-club of size exactly `. The claimed
fixed-parameter tractability with respect to n − `, however, only holds for the problem
of finding an s-club of size at least `. The other fixed-parameter tractability results
hold for both variants.
4.1.2 Further Structural Parameters
We extend the list of graph classes and properties described in Section 2.1
by some structural graph parameters that are only used in this chapter (see
Figure 4.2 for an illustration of their relations).
The minimum clique cover is the minimum number of cliques in a graph
that are needed to cover all vertices, that is, each vertex is contained in
at least one of these cliques. The bandwidth of a graph G = (V, E) is the
minimum k ∈ N such that there is a function f : V → N with |f (v)−f (u)| ≤ k
for all edges {u, v} ∈ E.
For a graph class Π (for example, bipartite graphs) the parameter distance
to Π measures the number of vertices that have to be deleted in the input
graph to transform it into a graph that is isomorphic to one in Π. If the
graph class Π is non-trivial3 and hereditary, then the problem to determine
the (minimum) parameter value distance to Π is NP-hard [LY80]. According
to the discussion in Section 2.3.5, we thus assume that a witness structure,
that is, a vertex subset whose deletion leads to a graph in Π, is always
provided as an additional input. However, we do not assume that it is a
minimum-size witness structure.
Note that the three parameters vertex cover, distance to cluster graphs,
and distance to cographs are special names for the parameters distance
to P2 /P3 /P4 -free graphs. Furthermore, s-clubs are exactly the connected
graphs not containing a Ps+2 on a shortest path. Hence, defining the class
of 2-club cluster graphs to contain all graphs not containing a P4 on a
shortest path, this is the natural extension of the “chain” formed by the
parameters distance to P2 /P3 /P4 -free graphs with respect to the stronger
parameterization relation (see Figure 4.2 for further relations). Finally, we
mention that deleting all the vertices on a Pt yields a factor-t approximation
for the parameter distance to Pt -free graphs (even restricted to Pt ’s on
shortest paths).
4.1.3 Our Contribution
We make progress towards a systematic classification of the complexity of
2-Club with respect to structural graph parameters. Figure 4.2 gives an
overview of our results and their implications.
In Section 4.2 we provide several NP-hardness results. Specifically, we
consider in Section 4.2.1 the graph parameters minimum clique cover number,
domination number, and some related graph parameters. Let G be the input
graph. We show that 2-Club is NP-hard even if the minimum clique cover
number of G is three. In contrast, we show that if the minimum clique
3A
graph class is non-trivial if it contains and excludes an infinite number of graphs.
79
cover number is two, then 2-Club is polynomial-time solvable. Then, we
show that 2-Club is NP-hard even if G has a dominating set of size two.
This result is tight in the sense that 2-Club is trivially solvable in case G
has a dominating set of size one. In Section 4.2.2, we study the parameter
distance to bipartite graphs. We show that 2-Club is NP-hard even if G can
be transformed into a bipartite graph by deleting only one vertex. This is
somewhat surprising since 2-Club is polynomial-time solvable on bipartite
graphs [Sch09].
Then, in Section 4.2.3, we consider the graph parameter h-index. The
study of this parameter is motivated by the fact that the h-index is usually
small in social networks (see Section 4.2.3 for a more detailed discussion).
On the positive side, we show that 2-Club is polynomial-time solvable for
constant h-index. On the negative side, we show that 2-Club parameterized
by the h-index of the input graph is W[1]-hard. Even worse, we prove that
2-Club is NP-hard even for degeneracy five. Note that degeneracy is at
most as large as the h-index of a graph.
In Section 4.3 we provide parameterized algorithms for parameter treewidth
and parameterizations that are related to the distance to “cluster-like graphs“.
More specifically, Section 4.3.1 contains a description of a parameterized
algorithm for the parameter distance to cographs and we show that it can
be slightly improved for the weaker parameter distance to cluster graphs.
Interestingly, these are rare examples for structural graph parameters which
are unrelated to treewidth and still admit a parameterized algorithm (see
Figure 4.2). Notably, the parameterized algorithm for treewidth and those for
k
distance to cograph both have basically the same running time, that is, 2O(2 ) ·
O(1)
n
, and this is so far also the best for the weaker parameter vertex cover.
Section 4.4 contains our findings concerning the kernelizability of 2-Club.
In Section 4.4, we give an O(k 2 )-vertex kernel for the parameter cluster
editing set and an O(k)-size kernel for the parameter feedback edge set.
The kernelization results for these rather large parameters are motivated by
our negative results: We show that 2-Club does not admit a polynomial
kernel with respect to a vertex cover of the underlying graph, unless NP ⊆
coNP/poly. This excludes polynomial kernels for many prominent structural
parameters such as feedback vertex set, pathwidth, and treewidth. For the
sake of completeness, we would like to mention that for the parameter
maximum degree, taking the disjoint union of the input graphs is a simple
OR-cross-composition algorithm (see Section 2.3.4) from 2-Club that proves
the non-existence of polynomial kernels, unless NP ⊆ coNP/poly.
0
A simple branching strategy shows that s-Club can be solved in O(2k ·nm)
0
time for the dual parameter k = n − ` [Sch09, Sch+12]: As long as there
is a vertex pair whose distance is at least three, branch into the case either
80
to delete the first or to delete the second vertex. In Section 4.5, we prove
that unless the SETH (Hypothesis 2.3.17) fails, s-Club cannot be solved in
0
O((2 − )k · |G|O(1) ) time for all > 0. This is evidence that Schäfer et al.’s
search tree algorithm is optimal with respect to the parameter k 0 . To prove
this, we give a reduction from Sat to 2-Club where k 0 in the resulting
instance is equal to the number of variables in the Sat-instance. Moreover,
the presented reduction also implies that s-Club does not admit a polynomial
kernel with respect to k 0 , unless NP ⊆ coNP/poly, answering an open
question by [Sch+12].
Having explored the limits of parameterized algorithmics for the dual
parameter k 0 on the theoretical side, in Section 4.6 we examine its usefulness
for solving 2-Club in practice. To this end, we implemented Schäfer et al.’s
search tree strategy for the dual parameter together with data reduction
rules that are partially deduced from our findings in Section 4.4. We explore
the effectiveness of our algorithm on random as well as on large-scale realworld graphs and show that our implementation outperforms all previously
implemented exact algorithms for 2-Club on random and on large-scale
real-world graphs. Especially on large graphs Schäfer et al.’s Turing kernelization for 2-Club turns out to be the most efficient technique in our
“parameterized toolbox”.
81
8
minimum
clique cover
8
maximum
independent set
8
domination
number
distance to clique
distance to
2-club [Sch+12]
distance to
2-club cluster
diameter
vertex cover
FPT, but no polynomial
kernels
8
max leaf number
feedback
edge set
cluster editing
FPT and polynomial kernels
distance to
disjoint paths
pathwidth
8
distance to
cluster
feedback
vertex set
8
8
distance to
interval
8
8
distance to
co-cluster
distance to
cograph
8
chromatic
number
treewidth
distance to
bipartite
8
distance to
chordal [BBT05]
distance to
perfect
NP-hard with constant parameter values
bandwidth
8
8
maximum
degree [Sch+12]
h-index
8
W[1]-hard
degeneracy
8
average
degree
Figure 4.2: Overview of the “stronger parameterization” relation between structural graph parameters and of our results
marked
for 2-Club. An edge from a parameter α to a parameter β below of α means that β is a stronger parameterization
than α (see Definition 2.3.14). The boxes indicate the complexity of 2-Club with respect to the enclosed parameters.
Specifically, the “orange” box (with parameter vertex cover on top) consists of all parameters for which 2-Club becomes
fixed-parameter tractable but does not admit a polynomial kernel unless NP ⊆ coNP/poly (Proposition 4.4.15, Theorem 4.4.16,
k
and Corollary 4.5.3). Therein, for all parameters the best performing algorithms run in 2O(2 ) · n time (see Theorem 4.3.6 for
0
parameter treewidth) with the only exception distance to 2-club admitting a 2k · nO(1) -time algorithm [Sch+12]. The “green”
box consisting of cluster editing, max leaf number, and feedback edge set contains all parameters admiting a single-exponential
time algorithm and a polynomial kernel (Theorems 4.4.5 & 4.4.14). The “red” box at the bottom contains those parameters
where 2-Club remains NP-hard even for constant values (see Section 4.2). It is open whether 2-Club is fixed-parameter
tractable when parameterized by distance to interval or distance to 2-club cluster and whether it admits a polynomial kernel
when parameterized by distance to clique.
82
4.1.4 Preliminaries: Twin Classes
We consider only connected input graphs (see Section 2.1 for a general
introduction to our graph-theoretic notation). Two vertices v and w are
twins if N (v) \ {w} = N (w) \ {v} and they are twins with respect to a vertex
set X if N (v) ∩ X = N (w) ∩ X. The twin relation is an equivalence relation;
the corresponding equivalence classes are called twin classes. The following
observation is easy to see and it shows that either none or all vertices of a
twin class are contained in a maximum-size s-club.
Observation 4.1.1. Let S ⊆ V be an s-club in a graph G = (V, E) and
let u, v ∈ V be twins. If u ∈ S and |S| > 1, then S ∪{v} is also an s-club in G.
In the hardness proofs provided in the next section, by adding “a huge
number” of twins we will often enforce that a certain vertex or one of its
twins has to be contained in the desired 2-club. Then, by Observation 4.1.1
we may assume that all of these twins are contained.
Remark. Note that one can check in O(nm) time whether a graph is an
s-Club by applying a breath-first search starting from each vertex.
4.2 NP- and W[1]-Hardness Results
In this section we provide several hardness proofs for parameters shaping the
“red box” in Figure 4.2. We also provide the W[1]-hardness proof for h-index.
4.2.1 Clique Cover Number and Domination Number
We prove that on graphs of diameter at most three, 2-Club is NP-hard even
if the minimum clique cover number is three or the domination number is
two. We first show that these bounds are tight. Towards this, observe that
the size of a maximum independent set is at most the size of a minimum
clique cover. Moreover, since each maximal independent set is a dominating
set, the domination number is also at most the size of a minimum clique
cover (see the “chain” on the left hand side of Figure 4.2).
Theorem 4.2.1. For s ≥ 2, s-Club can be solved in O(nm) time on graphs
where the size of a maximum independent set is at most two.
Proof. Let G = (V, E) be a graph. If a maximum independent set in G has
size one or G has diameter s, then V is an s-club. Moreover, a graph with a
maximum independent set size two is a 3-club, because one could select an
independent set of size three on a length-four shortest path. Hence we are
left with the case of 2-Club and G having a maximum size-two independent
83
set and diameter at least three. Thus there are two vertices v, u ∈ V with
dist(v, u) > 2. Since {v, u} is a maximum independent set and thus also a
dominating set, it follows that N [u] ∪ N [v] = V . Indeed, set N [v] (N [u]) is
a clique because two non-adjacent vertices in N (v) (N (u)) together with u
(with v) would form an independent set. We prove that N2 [v] or N2 [u] is a
maximum-size 2-club in G and thus can be determined in O(nm) time via
using a breath-first search starting from each vertex.
Let S be a maximum-size 2-club in G. First observe that either N [v] ⊆ S
or N [u] ⊆ S: Consider a vertex w ∈ N [u] such that S ∪ {w} is not a 2-club,
implying that in G[S ∪ {w}] w has distance at least three to some vertex
from N [v] ∩ S. Then N (w) ∩ N [v] ∩ S = ∅ and thus all vertices in N [u] ∩ S
have at least one neighbor in N [v] ∩ S, implying by the maximality of S that
N [v] ⊆ S. Since the argumentation is symmetric for w ∈ N [v], this proves
that v ∈ S or u ∈ S. Since N [v] and N [u] are cliques, it follows that N2 [v]
and N2 [u] are 2-clubs and, clearly, N2 [v] is the largest 2-club containing v
and, analogously, N2 [u] is the largest 2-club containing u.
The following theorem shows that the bound on the maximum independent
set size in Theorem 4.2.1 is tight.
Theorem 4.2.2. 2-Club is NP-hard on graphs with clique cover number
three and diameter three.
Proof. We describe a reduction from Clique. Let (G = (V, E), k) be a
Clique instance. We construct a graph G0 = (V 0 , E 0 ) consisting of three
disjoint vertex sets, that is, V 0 = V1 ∪V2 ∪VE . Further, for i ∈ {1, 2}, let Vi =
ViV ∪ Vibig , where ViV is a copy of V and Vibig is a set of n5 new vertices.
Let u, v ∈ V be two adjacent vertices in G and let u1 , v1 ∈ V1 , u2 , v2 ∈ V2
be the copies of u and v in G0 . Then add the vertices euv and evu to VE and
add the edges {v1 , evu }, {evu , u2 }, {u1 , euv }, {euv , v2 } to G0 . Furthermore,
3
add for each vertex v ∈ V the vertex set VEv = {e1v , e2v , . . . , env } to VE and
make v1 and v2 adjacent to all these new vertices. Finally, make the following
vertex sets to cliques: V1 , V2 , VE , and V1big ∪ V2big . Observe that G0 has
diameter three and that it has a clique cover number of three. We now
prove that
G has a clique of size k ⇔
k
G has a 2-club of size at least ` = 2n + kn + 2k + 2
.
2
0
5
3
“⇒”: Let S be a clique of size k in G. Let Sc contain all the copies of the
vertices of S. Furthermore, let SE = {euv | u1 ∈ Sc ∧ v2 ∈ Sc } and let Sb =
84
{eiv | v ∈ S ∧ 1 ≤ i ≤ n3 }. We now show that S 0 = Sc ∪ SE ∪ Sb ∪ V1big ∪ V2big
is a 2-club of size `. First, observe that |V1big ∪ V2big | = 2n5 and |Sc | = 2k.
Hence, |Sb | = kn3 and |SE | = 2 k2 . Thus, S 0 has the desired size. With a
straightforward case distinction one can check that S 0 is indeed a 2-club.
“⇐”: Let S 0 be a 2-club of size at least `. Observe that G0 consists
of |V 0 | = 2n5 + 2n + 2m + n4 vertices. Since ` > 2n5 at least one vertex
of V1big and at least one of V2big is in S. Since all vertices in V1big and
in V2big are twins, we can assume by Observation 4.1.1 that all vertices
of V1big ∪ V2big are contained in S 0 . Analogously, it follows that at least k
1
2
3
k
sets VEv , VEv , VEv , . . . , VEv are completely contained in S 0 . Since S 0 is a
j
2-club, the distance from vertices in Vibig to vertices in VEv is at most two.
j
j
Hence, for each set VEv in S 0 the two neighbors v1j and v2j of vertices in VEv are
j
j
also contained in S 0 . Since the distance of v1i and v2 for v1i , v2 ∈ S 0 is also at
most two, the vertices evi vj and evj vi are part of S 0 as well. Consequently, v i
and v j are adjacent in G. Therefore, the vertices v 1 , . . . , v k form a size-k
clique in G.
Since a maximum independent set is also a dominating set, Theorem 4.2.2
implies that 2-Club is NP-hard on graphs with domination number three
and diameter three. In contrast, for domination number one 2-Club is
trivial. The following theorem shows that this cannot be extended.
Theorem 4.2.3. 2-Club is NP-hard even on graphs with domination number two and diameter three.
Proof. We present a reduction from Clique. Let (G = (V, E), k) be a
Clique instance and assume that G is connected. We construct the graph G0
as follows. First copy all vertices of V into G0 . In G0 the vertex set V will
form an independent set. Now, for each edge {u, v} ∈ E add an edge-vertex
e{u,v} to G0 and make e{u,v} adjacent to u and v. Let VE denote the set of
edge-vertices. Next, add a vertex set C of size n + 2 to G0 and make C ∪ VE
a clique. Finally, add a new vertex v ∗ to G0 and make v ∗ adjacent to all
vertices in V . Observe that v ∗ plus an arbitrary vertex from VE ∪ C are a
dominating set of G0 and that G0 has diameter three. We complete the proof
by showing that
G has a clique of size k ⇔ G0 has a 2-club of size at least |C| + |VE | + k.
“⇒”: Let K be a size-k clique in G. Then, S = K ∪ C ∪ VE is a size|C| + |VE | + k 2-club in G: First, each vertex in C ∪ VE has distance two to
all other vertices S. Second, each pair of vertices u, v ∈ K is adjacent in G
and thus they have the common neighbor e{u,v} in VE .
85
“⇐”: Let S be a 2-club of size at least |C| + |VE | + k in G0 . Since
|C| > |V ∪ {v ∗ }|, it follows that there is at least one vertex c ∈ S ∩ C. Since c
and v ∗ have distance three, it follows that v ∗ 6∈ S. Now since S is a 2-club,
each pair of vertices u, v ∈ S ∩ V has at least one common neighbor in S.
Hence, VE contains the edge-vertex e{u,v} . Consequently, S ∩ V is a size-k
clique in G.
4.2.2 Distance to Bipartite Graphs
A 2-club in a bipartite graph is a biclique (a complete bipartite graph).
Finding a biclique with a maximum number of vertices can be done via
matching in bipartite graphs, hence 2-Club is polynomial-time solvable on
bipartite graphs [Sch09]. However, we show that 2-Club is already NP-hard
on graphs that become bipartite by deleting only one vertex.
Theorem 4.2.4. 2-Club is NP-hard even on graphs with distance one to
bipartite graphs.
Proof. We reduce from the NP-complete Maximum 2-Sat problem.
Maximum 2-Sat [GJ79](LO5)
Input: A boolean CNF formula F with two literals in each clause and
a positive integer k.
Question: Is there a truth assignment to the variables that satisfies at
least k clauses?
Given an instance F of Maximum 2-Sat where we assume that each
clause occurs only once, we construct an undirected graph G = (V, E). Let
C = {C1 , . . . , Cm } be the clauses of F over the variable set X = {x1 , . . . , xn }.
The vertex set V consists of the four disjoint vertex sets VC , VF , VX1 , VX2 ,
and one additional vertex v ∗ . The construction of the four subsets of V is as
follows.
The vertex set VC contains one vertex ci for each clause Ci ∈ C. The
5
vertex set VF contains for each variable x ∈ X exactly n5 vertices x1 , . . . , xn .
The vertex set VX1 contains for each variable x ∈ X two vertices: xt which
corresponds to assigning true to x and xf which corresponds to assigning false
to x. The vertex set VX2 is constructed similarly, but for every variable x ∈ X
3
it contains 2 · n3 vertices: the vertices x1t , . . . , xnt which correspond to
3
assigning true to x, and the vertices x1f , . . . , xnf which correspond to assigning
false to x.
Next, we describe the construction of the edge set E. The vertex v ∗
is made adjacent to all vertices in VC ∪ VF ∪ VX1 . Each vertex ci ∈ VC is
made adjacent to the two vertices in VX1 that correspond to the two literals
86
in Ci . Each vertex xi ∈ VF is made adjacent to xt and xf , that is, the
two vertices of VX1 that correspond to the two truth assignments for the
variable x. Finally, each vertex xit ∈ VX2 is made adjacent to all vertices
of VX1 except to the vertex xf . Similarly, each xif ∈ VX2 is made adjacent to
all vertices of VX1 except to xt . This completes the construction of G which
can clearly be performed in polynomial time. Observe that the removal of v ∗
makes G bipartite: each of the four vertex sets is an independent set and
the vertices of VC , VF , and VX2 are only adjacent to vertices of VX1 .
The main idea behind the construction is as follows. The size of the 2-club
forces the solution to contain the majority of the vertices in VF and VX2 .
As a consequence, for each x ∈ X exactly one of xt or xf is in the 2-club.
Hence, the vertices from VX2 in the 2-club represent a truth assignment. In
order to fulfill the bound on the 2-club size, at least k vertices from VC are
in the 2-club; these vertices can only be added if the corresponding clauses
are satisfied by the represented truth assignment. It remains to prove that
(C, k) is a yes-instance of Maximum 2-Sat ⇔
G has a 2-club of size n6 + n4 + n + k + 1.
“⇒”: Let β be an assignment for X that satisfies k clauses C1 , . . . , Ck of C.
Consider the vertex set S that consists of VF , v ∗ , the vertex set {c1 , . . . , ck } ⊆
VC that corresponds to the k satisfied clauses, and for each x ∈ X of
3
the vertex set {xt , x1t , . . . , xnt } ⊆ VX1 ∪ VX2 if β(x) = true and the vertex
3
set {xf , x1f , . . . , xnf } ∈ VX1 ∪ VX2 if β(x) = false. By the details of the
construction, |S| = n6 + n4 + n + k + 1. In the following, we show that S is
1
2
a 2-club. Herein, let SX
= VX1 ∩ S, SX
= VX2 ∩ S, and SC = VC ∩ S.
∗
1
First, v is adjacent to all vertices in SC ∪ VF ∪ SX
. Hence, all vertices
2
of S \ SX are within distance two in G[S]. By construction, the vertex
1
2
2
sets SX
and SX
form a complete bipartite graph in G: A vertex xit ∈ SX
1
1
is adjacent to all vertices in VX except xf which is not contained in SX .
2
2
The same argument applies to some xif ∈ SX
. Hence, the vertices of SX
are
1
2
neighbors of all vertices in SX . This also implies that the vertices of SX
∗
are in G[S] within distance two from v and from every vertex in VF since
1
each vertex of VF ∪ {v ∗ } has at least one neighbor in SX
. Finally, since
the k vertices in SC correspond to clauses that are satisfied by the truth
1
assignment β, each of these vertices has at least one neighbor in SX
. Hence,
2
every vertex in SX has in G[S] distance at most two to every vertex in SC .
1
2
“⇐”: Let S be a 2-club of size n6 +n4 +n+k+1, and let SX
= VX1 ∩S, SX
=
2
2
VX ∩ S, SF = VF ∩ S and SC = VC ∩ S. Clearly, neither SX = ∅ nor SF = ∅.
Since |VC | + |VX1 | + |VX2 | + 1 ≤ n2 + 2n + 2n4 + 1 < n5 for sufficiently
large n, S contains more than n6 − n5 vertices from VF . Consequently, for
each x ∈ X there is an index 1 ≤ i ≤ n5 such that xi ∈ SF .
87
We next show that for each x ∈ X it holds that either xt or xf is
1
contained in SX
. Towards this, since S is a 2-club, every vertex pair xi ∈ SF
2
and u ∈ SX has at least one common neighbor in S. By construction, this
1
common neighbor is a vertex of SX
and thus either xt or xf . Moreover, by
the observation above for each x ∈ X at least one xi is contained in SF .
1
Thus, for each x ∈ X at least one of xt and xf is contained in SX
.
1
2
1
Now observe that, G[SX ∪ SX ] is a complete bipartite graph, since SX
2
2
1
and SX are independent sets and SX has only neighbors in SX . This implies
that if for some x ∈ X there exist indices 1 ≤ i, j ≤ n3 with xit and xjf being
2
1
in SX
, then xt and xf are not in SX
. This contradicts the above observation
1
that at least one of xt and xf is in SX
. Moreover, since |VC | + |VX1 | + 1 ≤
2
3
4
2
n + 2n + 1 < n and |S \ VF | > n , we have |SX
| > n4 − n3 . It follows that
3
2
for each x ∈ X there is an index 1 ≤ i ≤ n such that either xit ∈ SX
or
i
2
1
xf ∈ SX . Altogether, this implies that either xt or xf is not contained in SX
.
6
4
In summary, S contains at most n vertices from VF , at most n vertices
2
1
from SX
, exactly n vertices from SX
, and thus there are k + 1 vertices
∗
2
in SC ∪ {v }. Since S is a 2-club that has nonempty SX
, every one of the
1
at least k vertices from SC has at least one neighbor in SX
. Because for
1
1
each x ∈ X either xf or xt is in SX , the n vertices from SX correspond to
an assignment β of X. By the above observation, this assignment satisfies
at least k clauses of C.
4.2.3 Parameterizing by the Sparsity: Average Degree,
h-index, and Degeneracy
2-Club is fixed-parameter tractable for the parameter maximum degree (the
algorithm of Schäfer et al. [Sch+12] can be analyzed in that way without
any changes). It has been observed that in large-scale biological [Jeo+00]
and social networks [BA99] the degree distribution often follows a power law,
implying that there are some high-degree vertices while most vertices have
low degree. This implies that these graphs are usually sparse, that is, the
number of edges is a small constant factor (average degree) in the number
of vertices. Compared to the maximum degree, this suggests considering
stronger parameters such as h-index, degeneracy, and average degree. For
any graph it holds that average degree ≤ 2 · degeneracy ≤ 2 · h-index, see
also Figure 4.2 for further relationships. For example, analyzing the coauthor
network derived from the DBLP dataset4 with more than 715 thousand
vertices, maximum degree 804, h-index 208, degeneracy 113, and average
degree 7 shows that also in real-world social networks these parameters
4 The
dataset and a corresponding documentation are available online (http://dblp.
uni-trier.de/xml/). Accessed Feb. 2012
88
are considerably smaller than the maximum degree (see Section 4.6 for an
analysis of these parameters on a broader dataset). In this section we show
that 2-Club is NP-hard even for constant small values of average degree and
degeneracy. In addition, we prove W[1]-hardness with respect to the h-index.
Unsurprisingly, 2-Club is NP-hard even with constant average degree.
Proposition 4.2.5. For any constant α > 2, 2-Club is NP-hard on connected graphs with average degree at most α.
Proof. Let (G, `) be an instance of 2-Club where ∆ is the maximum degree
of G. We can assume that ` > ∆ + 2 since, as shown for example in the proof
of Theorem 4.2.2, 2-Club remains NP-hard in this case. We add a path P
to G and an edge from an endpoint p of P to an arbitrary vertex v ∈ V .
Since ` > ∆ + 2, any 2-club of size at least ` contains at least one vertex
that is not in P . Furthermore, it cannot contain p and v since in this case
it is a subset of either N [v] or N [p] which both have size at most ∆ + 2 (v
has degree at most ∆ in G). Hence, the instances are equivalent. Putting at
2m
− ne vertices in P ensures that the resulting graph has average
least d α−2
degree at most α.
We remark that the bound in Proposition 4.2.5 is tight:PConsider a connectedP
graph G with average degree at most two, that is, n1 v∈V deg(v) ≤ 2.
Since v∈V deg(v) = 2m, it follows that n ≥ m and, thus, the feedback
edge set of G contains at most one edge. As 2-Club is fixed-parameter
tractable with respect to feedback edge set (Section 4.4), it follows that
2-Club can be solved in polynomial time on connected graphs with average
degree at most two.
Proposition 4.2.5 suggests considering weaker parameters such as degeneracy or h-index of G (see Figure 4.2). Recall that having h-index k means
that there are at most k vertices with degree greater than k. Since social
networks typically have small h-index, fixed-parameter tractability with
respect to the h-index would be desirable. Unfortunately, 2-Club turns out
to be W[1]-hard when parameterized by the h-index and NP-hard on graphs
of constant degeneracy. Following the W[1]-hardness result, we show that
there is “at least” an XP-algorithm (Definition 2.3.3) implying that 2-Club
is polynomial-time solvable for constant h-index.
We reduce from the W[1]-hard Multicolored Clique problem [Fel+09].
Multicolored Clique
Input: An undirected graph G = (V, E) and a proper k-coloring c
of G.
Question: Is there a multicolored clique of size k in G, that is, a clique C ⊆
V such that c(v) 6= c(v 0 ) for all v, v 0 ∈ C with v 6= v 0 ?
89
Lemma 4.2.6. There are two polynomial-time computable reductions that
compute for any instance (G, c, k) of Multicolored Clique an equivalent
2-Club-instance (G0 , `) such that G0 has diameter three and, additionally,
in reduction i) G0 has h-index at most k + 7 and in reduction ii) G0 has
degeneracy five.
Proof. The only difference between both reductions is the construction
of a so-called coloring gadget. We first describe the common part. Let
(G, c, k) with G = (V, E) and c : V → {1, . . . , k} be an instance of Multicolored Clique. We construct a graph G0 and choose ` ∈ N such
that (G0 , `) is a yes-instance for 2-Club if and only if (G, c, k) is a yesinstance for Multicolored Clique. We will first construct some structures in G0 which allow to describe the basic ideas: For each vertex v ∈ V
create a vertex gadget by adding the α-vertices {α1v , . . . , αnv }, the β-vertices
v
{β1v , . . . , βn+1
}, and the γ-vertices {γ1v , . . . , γnv }, and {ωαv , ωγv }. Add edges
v
v
such that (α1 , β1v , γ1v , α2v , β2v , γ2v , . . . , αnv , βnv , γnv , ωαv , βn+1
, ωγv , α1v ) induces a
cycle. Add the three vertices
U = {uα , uβ , uγ }
and add edges from all α- (β-,γ-)vertices to uα (uβ , uγ ), respectively. Add
the edges {ωαv , uα } and {ωγv , uγ }. Furthermore, for a fixed ordering V =
{v1 , . . . , vn } add for each edge {vi , vj } ∈ E an edge-vertex evi ,vj that is adv
jacent to each of {αjvi , βjvi , γi j }. (Observe that the α- and γ-vertex neighbor
are in different vertex gadgets.) The following property is fulfilled:
1. For each vertex v in a vertex gadget it holds that |N (v) ∩ U | = 1 and
for each u ∈ U \ N (u) it holds that |N (v) ∩ N (u)| = 1 .
The idea of the construction is that U will be forced to be contained in any
2-club S of size at least `. Hence by Property 1 it follows that if an α-vertex
in a vertex gadget is contained in S, then the unique β- and γ-vertex in
its neighborhood have to be contained in S as well. Since this argument
symmetrically holds for β- and γ-vertices, it follows that either all or none
of the vertices from a vertex gadget are contained in S. Observe that, in
this context, ωαv (ωγv ) behaves like a “normal” α- (γ-) vertex. Analogously,
each edge-vertex evi ,vj ∈ S needs to have a common neighbor with each
vertex of U . Thus, evi ,vj ∈ S implies that all vertices in the two vertex
gadgets that correspond to vi and vj are contained in S. By connecting the
vertices {ωαv , ωγv } appropriately we will ensure that for each color c at most
one vertex gadget whose vertex in G is colored with c can have a non-empty
intersection with S. (The construction of the corresponding coloring gadget
is the only part where the two reductions differ.) Furthermore, we choose the
90
value of` such that S contains vertices from at least k vertex gadgets and at
least k2 edge-vertices. Hence, there are exactly k vertex gadgets together
with k2 edge-vertices that contribute to S. Since the vertices corresponding
to the vertex gadgets have different
colors and since the endpoints of the
edges corresponding to the k2 edge-vertices are all within this set of k
vertices, the set S corresponds to a multicolored clique in G.
To complete the construction and to ensure the properties discussed above,
we next add the anchor gadget and the coloring gadget. To argue about their
correctness we claim that, eventually,
|V 0 | =
n(3n + 3) + 4n3 + 7 +
| {z }
| {z }
n vertex gadgets
anchor gadget
m
|{z}
+
m edge-vertices
|VC |
|{z}
(4.1)
|VC |.
| {z }
(4.2)
coloring gadget
and we set
`=
3
k(3n + 3) + 4n + 7 +
| {z }
| {z }
k vertex gadgets
anchor gadget
k
2
| {z }
+
coloring gadget
(k2) edge-vertices
Anchor Gadget: We denote by VA the set of all vertices in the anchor
gadget including U . Finally, there will be 4n3 + 7 vertices in VA . Besides U
the anchor gadget will contain only four other vertices, namely {lU , l, r1 , r2 }
that have neighbors outside the gadget. Before describing the construction we
will list some of its properties that will be used in the argumentation later on.
2. The set U is contained in any 2-club in G0 of size at least `.
3. A 2-club of size at least ` contains either all or none of the vertices of
a vertex gadget.
4. For any two vertices u ∈ U and v ∈ {r1 , r2 } it holds that (N (u) ∪
N (v)) ∩ VA = VA \ (U ∪ {v}), that (N (l) ∪ N (v)) ∩ VA = VA \ U , and
that (N (lU ) ∪ N (v)) ∩ VA = VA .
Informally, Property 4 ensures that if a vertex is adjacent to one of
{lU , l} ∪ U and to one of {r1 , r2 }, then it has distance at most two to all
vertices in VA \ U and also distance at most two to all of U if it is adjacent
to lU .
The anchor gadget is constructed as follows (see Figure 4.3): Add four sets
Vα , Vβ , Vγ , Vα,β,γ each of size n3 , and add edges from each vertex in Vα,β,γ
to each one in U ∪ {l, lU }. Additionally, add edges from each vertex in Vα
to each of {uα , r1 , r2 }, from each vertex in Vβ to each of {uβ , r1 , r2 }, and
from each vertex in Vγ to each of {uγ , r1 , r2 }. Finally, add edges such that
{lU , l, r1 , r2 } is a clique and an edge from lU to each vertex in U .
91
uα
uβ
Vα
uγ
Vα,β,γ
Vβ
lU
r1
l
r2
Vγ
Figure 4.3: The anchor gadget. The vertices {uα , uβ , uγ , lU , l, r1 , r2 } are the only
vertices which have neighbors outside the anchor gadget. All the vertices in the
sets Vα,β,γ , Vα , Vβ , and Vγ are twins and uα (uβ , uγ ) is the only common neighbor
between Vα,β,γ and Vα (Vβ , Vγ , resp.).
By the construction above, Property 4 is fulfilled and the anchor gadget
is a 2-club. Observe that uα (uβ , uγ ) is the only common neighbor of any
vertex in Vα,β,γ and of any vertex in Vα (Vβ , Vγ , resp.). Hence, if at least one
vertex from each set Vα , Vβ , Vγ , Vα,β,γ is contained in a 2-club, then also U
is contained in it. To prove Property 2, let S ⊆ V 0 be a 2-club of size ` that
is disjoint to at least one of {Vα , Vβ , Vγ , Vα,β,γ }. The number of vertices that
are not in S is at most |V 0 | − `, which is (see Equations (4.1) & (4.2)):
k
0
|V | − ` = (n − k)(3n + 3) + m −
< n3 .
2
This yields a contradiction and proves Property 2. As argued above, Properties 1 & 2 imply the correctness of Property 3.
Recall that so far only U has neighbors outside the anchor gadget, namely
all α- (β-,γ-)vertices are adjacent to uα (uβ , uγ , resp.) and ωαv (ωγv ) is
adjacent to uα (uγ ). We describe via properties how to connect the anchor
gadget to the vertex gadgets.
5. ωαv is adjacent to r1 and ωγv is adjacent to r2 for all v ∈ V .
6. All α-, β-, and γ-vertices and all edge-vertices are adjacent to each of
{r1 , r2 }. Additionally, each edge-vertex is adjacent to l.
92
Observe that Property 6 does not violate the correctness of Property 3 since
the vertices r1 , r2 are not neighbors of any vertex in U (see Property 4).
0
Altogether, Properties 4 to 6 imply that all vertex pairs in G0 except {ωαv , ωγv }
with v 6= v 0 have distance at most two.
We next construct the coloring gadget that guarantees that only those
0
vertex pairs {ωαv , ωγv } have a common neighbor (and thus can be contained
in any 2-club) for which c(v) 6= c(v 0 ). We will give two different constructions
of the coloring gadget where the first guarantees an h-index of at most k + 7
and the second guarantees degeneracy five. Denoting the set of vertices in
the coloring gadget by VC , both constructions fulfill the following properties:
7. Each vertex in VC is adjacent to each of {lU , r1 , r2 }.
0
8. Any pair {ωαv , ωγv }, v =
6 v 0 , has a common neighbor in VC if and only
if c(v) 6= c(v 0 ).
The two properties above are sufficient to prove the correctness of both
reductions. We next provide the two constructions.
Coloring gadget i): For each color i ∈ {1, . . . , k} add a vertex ci and
let VC = {c1 , . . . , ck } be the vertex set containing these vertices. Add an
edge between a vertex ωαv and ci if c(v) = i and an edge from ωγv to ci if
c(v) 6= i (Property 8). Finally, add edges such that each vertex in VC is
adjacent to each vertex in {lU , r1 , r2 } (Property 7).
Note that the h-index of G0 is at most |VC | + |U | + |{lU , l, r1 , r2 }| = k + 7,
as the vertices in VC ∪ U ∪ {lU , l, r1 , r2 } are the only ones that might have
degree at least k + 7.
0
Coloring gadget ii): For each pair {ωαv , ωγv } with c(v) 6= c(v 0 ) add
0
a vertex cv,v0 that is adjacent to each of {ωαv , ωγv } (Property 8). Finally,
denoting all these new vertices by VC we add an edge from each vertex in VC
to each vertex in {lU , r1 , r2 } (Property 7).
We next prove that G0 has degeneracy five by giving an elimination
ordering, that is, an order of how to delete vertices of degree at most five
that leads to an empty graph: In the anchor gadget each of the vertices in
Vα , Vβ , Vγ , Vα,β,γ has maximum degree five and hence they can be deleted.
Then, delete all vertices in VC , as each of them also has degree five. Delete
all edge-vertices (they also have degree five). In the remaining graph each
vertex in a vertex gadget (see Property 6) is adjacent to its two neighbors in
its vertex gadget, adjacent to one of U , and one or two neighbors in {r1 , r2 }.
Hence, all vertices in vertex gadgets can be removed as they have degree at
most five. The remaining vertices are U ∪ {lU , l, r1 , r2 } and all vertices in
U ∪ {l, r1 , r2 } have maximum degree four.
93
It remains to prove the correctness of the two reductions, that is,
(G, c, k) is a yes-instance of Multicolored Clique ⇔
(G0 , `) is a yes-instance of 2-Club.
“⇒”: Let C be a multicolored clique in G of size k. We construct a set S ⊆ V 0
of size ` and prove that it is a 2-club in G0 . The set S contains each vertex
gadget that corresponds to some vertex in C, the coloring gadget, the anchor
gadget, and any edge vertex evi ,vj with vi , vj ∈ C. See Equation (4.2) to
verify that |S| = `. To verify that S is a 2-club, note that for each vertex v
in a vertex gadget it holds that its unique common neighbor with any vertex
in U \ N (v) is contained in S and thus from Properties 1 & 4 to 6 it follows
that in G0 [S] the vertex v has distance at most two to any anchor gadget
vertex. Additionally, Properties 5 to 8 imply that v has distance at most
two to all other vertex gadget vertices in S, all coloring gadget vertices,
and all edge vertices in S. Properties 4, 6 and 7 imply that any coloring
gadget vertex has distance at most two to all anchor vertices, coloring gadget
vertices, and edge vertices. Finally, Properties 4 & 6 show that each edge
vertex has distance two to all anchor vertices.
“⇐”: Let S be a 2-club of size at least `. By Property 2 it follows
that U ⊆ S and by Property 3 it follows that each vertex gadget is either
fully contained in S or is disjoint to S. Denote by C the vertices in G
that correspond to the vertex gadgets that are fully contained in S. First,
0
since two vertices ωαv and ωγv , v =
6 v 0 , do not have a common neighbor if
0
c(v) = c(v ) (Property 8) and there are only k colors, it follows that |C|
≤ k.
Hence by Equations (4.1) & (4.2) it follows that S contains at least k2 edge
vertices. Since each edge vertex evi ,vj needs to have a common neighbor
with each vertex in U and the α- and the γ- vertex neighbors of evi ,vj are
in different vertex gadgets, it follows that {vi , vj } ⊆ C. From this, since
|C| ≤ k it follows that |C| = k and that S contains exactly k2 edge vertices,
implying that |C| induces a clique in G. Finally, note that this clique is
multicolored because of Property 8.
Lemma 4.2.6 has several consequences.
Corollary 4.2.7. 2-Club is NP-hard on graphs with degeneracy five.
Corollary 4.2.8. 2-Club parameterized by h-index is W[1]-hard.
Since the reduction in Lemma 4.2.6 is from Multicolored Clique
and in the reduction the new parameter is linearly bounded in the old one
(it is a linear-parameter transformation, see Definition 2.3.12), the results
of Chen et al. [Che+05] imply the following.
94
Corollary 4.2.9. 2-Club on graphs with h-index k cannot be solved in no(k) time unless the ETH fails.
We next prove that there is an XP-algorithm for the parameter h-index.
Therein, we mainly exploit the fact that if a graph G has h-index k, then
there is a set X of at most k vertices such that G − X has maximum degree
at most k. Since 2-Club is fixed-parameter tractable with respect to the
maximum degree, one can find largest 2-clubs in the connected components
of G − X and among them one has to ensure that they share common
neighbors in X.
4
k
Theorem 4.2.10. 2-Club can be solved in O(2k · n2 · n2 m) time where k
is the h-index of the input graph.
Proof. We give an algorithm that finds a maximum 2-club in a graph G0 =
4
k
(V 0 , E 0 ) in O(2k · n2 · n2 m) time where k denotes the h-index of G0 . Let
X 0 ⊆ V 0 be the set of all vertices in G0 with degree greater than k. By
definition of the h-index, |X 0 | ≤ k. For the proof of correctness fix any
maximum 2-club S in G0 . Throughout the algorithm via branching we will
guess some vertices contained in S and we will collect them in the set P .
Then, cleaning the graph means to exhaustively remove all vertices that do
not have distance at most two to all vertices in P . These vertices cannot be
contained in S and, clearly, if this requires to delete some vertex in P we
will abort this branch.
First, branch into the at most 2k cases to guess the set X = X 0 ∩ S
(potentially X = ∅). Delete all vertices from X 0 \ X, initialize P with X,
and clean the graph. Denoting the resulting graph by G = (V, E), we next
describe how to find a maximum 2-club in G that contains X. Towards this,
consider the at most 2k twin classes of the vertices in V \ X with respect
k
to X. Branch into the O(n2 ) cases to guess for each twin class T any
vertex from T ∩ S, called the center of T . Clearly, if T ∩ S = ∅, then there
is no center and we delete all vertices in T . Add all the centers to P and
clean the graph.
Two twin classes T and T 0 are in conflict if N G (T ) ∩ N G (T 0 ) ∩ X = ∅.
Now, the crucial observation is that, if T and T 0 are in conflict, then all
vertices in (T ∪ T 0 ) ∩ S are contained in the same connected component
of G[S \ X], since otherwise they would not have pairwise distance at most
two. However, this implies that all vertices in T ∩ S have pairwise distance
at most four in G[S \ X]. Hence, for each twin class T with center c that is
in conflict to any other twin class it holds that T ∩ S ⊆ N4G−X [c] and since
G − X has maximum degree at most k, one can guess N4S [c] = N4G−X [c] ∩ S
4
by branching into at most 2k cases. Delete all vertices in T guessed to be
S
not contained in N4 [c], add N4S [c] to P , and clean the graph. Note that the
95
remaining graph is a 2-club, since P contains X and the intersection of S
with each twin class that is in conflict to any other twin class. By definition
of twin classes that are in conflict, it holds that all other twin classes share
a common neighbor in X.
4.3 Fixed-Parameter Tractability Results
4.3.1 Distance to (Co-)Cluster Graphs and Cographs
In this section we present parameterized algorithms for 2-Club parameterized by distance to co-cluster graphs, by distance to cluster graphs, and by
k
distance to cographs. All these algorithms have running time 2O(2 ) · nO(1)
which is similar to the one obtained for treewidth (Theorem 4.3.6). For the
weaker parameters the constants in the exponential part of the running time
are smaller. Hence, none of the algorithms “dominates” one of the other
algorithms even with distance to cographs being a provably smaller parameter than distance to cluster graphs or distance to co-cluster graphs (see
Figure 4.2). As already mentioned, even for the considerably weaker paramk
eter vertex cover the best known algorithm has running time 2O(2 ) · nO(1) .
In contrast, the parameter distance to clique which is unrelated to vertex
cover admits a trivial O(2k · nm)-time algorithm, even in case of the general
0
s-Club. This is implied by the O(2k · nm)-time algorithm for the dual
parameter k 0 = n − ` [Sch+12] which can be interpreted as distance to
2-clubs and the fact that each clique is a 2-club.
Distance to Co-Cluster Graphs and Distance to Cluster Graphs. To complete the picture drawn in Figure 4.2, we also give short description of an
algorithm for 2-Club parameterized by the distance to co-cluster graphs.
k
Theorem 4.3.1. 2-Club is solvable in O(2k ·22 ·nm) time where k denotes
the distance to co-cluster graphs.
Proof. Let (G, X, `) be an 2-Club instance where X has |X| = k and G−X is
a co-cluster graph. Note that the co-cluster graph G−X is either a connected
graph or an independent set. In the case that G − X is an independent set,
the set X is a vertex cover and we thus apply the parameterized algorithm
with respect to parameter vertex cover (see Theorem 4.3.4) to solve the
k
instance in O(2k · 22 · nm) time.
Hence, assume that G − X is connected. Since G − X is the complement of
a cluster graph, this implies that G − X is a 2-club. Thus, if ` ≤ n − k, then
we can trivially answer yes. Hence, assume that ` > n − k or, equivalently,
k > n − `. Schäfer et al. [Sch+12] showed that 2-Club can be solved
96
in O(2n−` nm) time (simply choose a vertex pair having distance at least
three and branch into the two cases of deleting one of them). Since k > n − `
it follows that 2-club can be solved in O(2k nm) time in this case.
Next, we present a parameterized algorithm for the parameter distance to
cluster graphs.
k
Theorem 4.3.2. 2-Club is solvable in O(2k ·32 ·nm) time where k denotes
distance to cluster graphs.
Proof. Let (G, X, `) be a 2-Club instance where G−X is a cluster graph and
|X| = k. First, branch into all possibilities to choose the subset X 0 ⊆ X that
is contained in the desired 2-club S. Then, remove X \ X 0 and all vertices
that are not within distance two to all vertices in X 0 , and let G0 = (V 0 , E 0 )
denote the resulting graph.
Let T = T1 , . . . , Tp be the set of twin classes of V 0 \ X 0 with respect to X 0
and let C1 , . . . , Cq denote the clusters of G0 − X 0 . Two twin classes T and T 0
are in conflict if N (T ) ∩ N (T 0 ) ∩ X 0 = ∅. The three main observations
exploited in the algorithm are the following:
1. If two twin classes Ti and Tj are in conflict, then all vertices of Ti that
are in a 2-club and all vertices from Tj that are in a 2-club must be in
the same cluster of G0 − X 0 .
2. Every vertex from G0 − X 0 can reach all vertices in X 0 only via vertices
of X 0 or via vertices in its own cluster.
3. If one 2-club vertex v ∈ S is in a twin class Ti and in a cluster Cj ,
then all vertices that are in Ti and in Cj can be added to S without
violating the 2-club property.
We exploit the above observations in a dynamic programming algorithm.
In this algorithm, we create a two-dimensional
table A where an entry A[i, T 0 ]
S
stores the maximum size of a set Y ⊆ 1≤j≤i Cj such that the twin classes
of Y are exactly T 0 ⊆ T and all vertices in Y have in G[Y ∪ X 0 ] distance at
most two to each vertex from Y ∪ X 0 .
Before filling the table A, we calculate a value s(i, T 0 ) that stores the
maximum number of vertices we can add from Ci that are from the twin
classes in T 0 and fulfill the requirements in the previous paragraph. This
0
value is defined as follows. Let CiT denote the maximal subset of vertices
0
0
from Ci whose twin classes are exactly T 0 . Then, s(i, T 0 ) = |CiT | if CiT
0
exists and every pair of non-adjacent vertices from CiT and from X 0 have
0
a common neighbor. Otherwise, set s(i, T ) = −∞. Note that as a special
case we set s(i, ∅) = 0. Furthermore, for two subsets T 00 and T̃ define
the predicate conf(T 00 , T̃ ) as true if there is a pair of twin classes Ti ∈ T 00
and Tj ∈ T̃ such that Ti and Tj are in conflict, and as false, otherwise.
97
Using these values, we now fill A with the following recurrence: We set
A[i, T 0 ] to
(
A[i − 1, T̃ ] + s(i, T 00 ) if T̃ ∪ T 00 = T 0 ∧ ¬ conf(T̃ , T 00 ),
max
otherwise.
T 00 ⊆T 0 ,T̃ ⊆T 0 −∞
This recurrence considers all cases of combining a set Y for the clusters C1
to Ci−1 with a solution Y 0 for the cluster Ci . Herein, a positive table entry is
only obtained when the twin classes of Y ∪ Y 0 is exactly T 0 and the pairwise
distances between Y ∪ Y 0 and Y ∪ Y 0 ∪ X 0 in G[Y ∪ Y 0 ∪ X 0 ] are at most
two. The latter property is ensured by the definition of the s() values and
by the fact that we consider only combinations that do not put conflicting
twin classes in different clusters.
Now, the table entry A[q, T 0 ] contains the size of a maximum vertex
set Y such that in G0 [Y ∪ X 0 ] every vertex from Y has distance two to all
other vertices. It remains to ensure that the vertices from X 0 are within
distance two from each other. This can be done by only considering a
table entry A[q, T 0 ] if each non-adjacent vertex pair x, x0 ∈ X 0 has either a
common neighbor in X 0 or in one twin class contained in T 0 . The maximum
size of a 2-club in G0 is then the maximum value of all table entries that
fulfill this condition.
k
The running time can be bounded by O(2k · 32 · nm): We try all 2k
partitions of X and for each of these partitions, we fill a dynamic programk
ming table with 22 · n entries. The number of overall table lookups and
k
k
updates is O(32 · n) since there are 32 possibilities to partition T into the
three sets T 00 , T̃ , and T \ T 0 . Since each Ci is a clique, the entry s(i, T 0 ) is
computable in O(nm) time and the overall running time follows.
Distance to Cographs. We proceed by describing the parameterized algorithm with respect to the parameter distance to cographs. Recall that
since cographs are exactly the P4 -free graphs, any connected component of
a cograph is a 2-club.
k
Theorem 4.3.3. 2-Club is solvable in O(2k · 82 · n4 ) time where k denotes
the distance to cographs.
Proof. Let G0 be the input graph of a 2-Club instance. Moreover, let X 0 be
a vertex subset of size at most k of G0 whose deletion results in a cograph.
We next describe a parameterized algorithm with respect to k = |X 0 | that
finds a maximum-size 2-club in G. For our correctness proof we fix any
maximum 2-club S in G0 . First branch into the at most 2k cases to guess
X = X 0 ∩ S. Delete all vertices in X 0 \ X. Denoting by G = (V, E) the
98
remaining graph, observe that G − X is a cograph. The remaining task is to
find a maximum 2-club in G that contains X.
Before proceeding to describe the algorithm we introduce the following
characterization of cographs [BLS99]: A graph is a cograph if it can be
constructed from single vertex graphs by a sequence of parallel and series
compositions. Given t vertex
St disjoint
St graphs Gi = (Vi , Ei ), the series
composition is the graph ( i=1 Vi , i=1 Ei ∪ {{v, u} | v ∈ Gi ∧ u ∈ Gj ∧
St
St
1 ≤ i < j ≤ t} and the parallel composition is ( i=1 Vi , i=1 Ei ). The
corresponding cotree of a cograph G is the tree whose leaves correspond
to the vertices in G and each inner node represents a series or parallel
composition of its children up to a root which represents G.
We next describe a dynamic programming algorithm that proceeds in a
bottom-up manner on the cotree of G − X and finds a maximum 2-club
in G that contains X. We may assume that t = 2 for all series and parallel
compositions, as otherwise we can simply split up the corresponding nodes
in the cotree. For each node P in the cotree let V (P ) ⊆ V \ X be the
vertices corresponding to the leaves of the subtree rooted in P . Furthermore,
consider the (at most 2k many) twin classesSof V \ X with respect to X and
for a subset of twin classes T let V (T ) = T ∈T T denote the union of all
vertices in the twin classes of T . We compute a table Γ where for any subset
of twin classes T and any node P of the cotree the entry Γ(P, T ) is the size
of a largest set L ⊆ V (P ) ∩ V (T ) that fulfills the following properties:
1. for all T ∈ T : T ∩ L 6= ∅ and
2. for all v ∈ L ∪ X and u ∈ L: distG[L∪X] (u, v) ≤ 2.
The intention of the definition of the table entries is that the graph G[L ∪ X]
is a “2-club-like” structure that contains a vertex from each twin class in T
(Item 1) and for any pair of vertices, except those where both vertices are
from X, have distance at most two (Item 2). Denoting the root of the
cotree by r and by Ts the set of all twin classes that have a non-empty
intersection with S, Γ(r, Ts ) ≥ |S \ X| as S \ X trivially fulfills all properties.
Reversely, for any subset of twin class T that contains for each pair of vertices
{u, v} ∈ X with distG[X] (u, v) > 2 a twin class T ∈ T with {u, v} ⊆ N (T ),
any set corresponding to Γ(r, T ) forms together with X a 2-club.
We now describe the dynamic programming algorithm. Let P be a leaf
node of the cotree with V (P ) = {x} and let T be any subset of twin
classes. The two sets {x} and ∅ are the only candidates for L. Hence we
set Γ(P, T ) = 1 if x fulfills both properties, Γ(P, ∅) = 0 (∅ fulfills both
properties), and Γ(P, T ) = −∞ otherwise. Next we describe the dynamic
programming algorithm for inner nodes of the cotree. Let P be any node of
the cotree with children P1 , P2 , and let T be any subset of twin classes. We
construct a graph GP by exhaustively deleting in G[(V (P ) ∩ V (T )) ∪ X] all
99
vertices from V (P ) ∩ V (T ) that have distance more than two to any vertex
in X. (Clearly, such a vertex has to be deleted because of Item 2.) If the
resulting graph GP violates Item 1, then there is no set corresponding to
Γ(P, T ) and thus we set the entry to be −∞. Additionally, if GP fulfills
all properties, then set Γ(P, T ) = |V (GP )| − |X|. To handle the remaining
case where GP violates only Item 2 we make a case distinction on the
node type of P .
Case 1: P is a series node.
Let {u, v} ⊆ V (GP ) \ X be a vertex pair with distGP (u, v) > 2. Since a
series composition introduces an edge between each vertex in V (P1 ) and
each vertex in V (P2 ) and V (GP ) ⊆ V (P ) = V (P1 ) ∪ V (P2 ), it follows
that either V (GP ) ∩ V (P1 ) = ∅ or V (GP ) ∩ V (P2 ) = ∅. This implies that
Γ(P, T ) = max{Γ(P1 , T ), Γ(P2 , T )}.
Case 2: P is a parallel node.
Consider any set L that corresponds to Γ(P, T ). By the definition of a
parallel node there is no edge between a vertex from V (P1 ) to a vertex
in V (P2 ). Consequently, any pair of vertices in L with one vertex in V (P1 )
and the other in V (P2 ) have a common neighbor in X. Correspondingly, we
say that two twin classes are consistent if they have at least one common
neighbor in X and two sets of twin classes are consistent if any twin class
of the first set is consistent with any twin class of the second set. Denoting
by T1S (T2S ) the set of twin classes with a non-empty intersection with
L ∩ V (P1 ) (L ∩ V (P2 )), by the argumentation above it follows that T1S
is consistent with T2S . Additionally, it is straightforward to verify that
L ∩ V (P1 ) (L ∩ V (P2 )) fulfills all properties (except being a largest set) for
the entry Γ(T1S , P1 ) (Γ(P2 , T2S )).
Reversely, for any two consistent sets of twin classes T1 , T2 let L1 (L2 ) be
any vertex set that corresponds to Γ(P1 , T1 ) (Γ(P2 , T2 )). It holds that L1 ∪L2
fulfills all properties for Γ(P, T1 ∪ T2 ) and hence Γ(P, T̃ ∪ T2 ) ≥ |L1 ∪ L2 |.
Hence it is correct to set Γ(T , P ) to be the largest value of Γ(P1 , T1 )+Γ(P2 , T2 )
where T1 , T2 are consistent and T1 ∪ T2 = T . This completes the description
of the algorithm.
k
The table Γ contains O(n · 22 ) entries as there are at most 2k twin classes.
k
Each entry can be computed in O(n3 + 42 ) time. In total, together with
the factor of 2k which is needed to guess X, the running time of the above
k
algorithm is O(2k · 82 · n4 ).
4.3.2 Treewidth
In this section, we show that 2-Club is fixed-parameter tractable when
parameterized by treewidth of the input graph. This was already shown
100
by Schäfer [Sch09] via a monadic second-order logic formulation, however,
here we present a direct combinatorial algorithm. Surprisingly, up to some
constants in the exponent, this is currently even the best algorithm for much
larger parameters such as vertex cover. To demonstrate the main ideas
behind the algorithm, we first describe its principles via an algorithm for
the weaker vertex cover parameter.
k
Theorem 4.3.4. s-Club is solvable in O(2k ·22 ·nm) time where k denotes
the size of a vertex cover.
Proof. Let (G = (V, E), X, s, `) be an s-Club instance where X is a vertex
cover of G with |X| = k . First, branch into all possibilities to choose the subset X 0 ⊆ X that is contained in the desired s-club. Then, remove X \ X 0 and
all vertices that are not within distance s to all vertices in X 0 . Clearly, V \ X
forms an independent set. Moreover, by Observation 4.1.1, it follows that two
vertices u, v ∈ V \ X that are twins with respect to X 0 are either both con0
tained in a maximal s-club or none of them. Since there are at most 2|X | ≤ 2k
different twins, branching into all possibilities to add them to the s-club takes
k
22 time. Finally, check in O(nm) time whether the resulting graph forms
k
an s-club. In total, s-Club can be solved in O(2k · 22 · nm) time.
Extending the ideas behind Theorem 4.3.4, we now give a direct combinatorial algorithm for the parameter treewidth. More specifically, we
provide a dynamic programming algorithm on a nice tree decomposition of
the input graph G (see Niedermeier [Nie06, Chapter 10] for more details
about nice tree decompositions): It is a tree decomposition
T , that is, a
S
tree with vertices X1 , . . . , Xr called bags such that i Xi = V (G) and for
each {u, v} ∈ E(G) there is a bag Xi with {u, v} ∈ Xi . Additionally, for
each v ∈ V (G) the bags containing v induce a connected component in T . In
a nice tree decomposition, each bag Xi is either an join node (it has exactly
two children with Xi = Xj = Xl ), an insert node (|Xi \ Xl | = 1 for the
only child Xl ), a forget node (|Xl \ Xi | = 1 for the only child Xl ), or a leaf
node (no children). An arbitrary tree decomposition can be restructured
in linear time into a nice tree decomposition without an increase of the
treewidth [Klo94], that is, the size of the largest bag minus one. Deciding
whether a graph has treewidth ω and (in case of its existence) constructing
3
a corresponding tree decomposition can be done in 2O(ω ) · n time [Bod96].
Lemma 4.3.5. Let G = (V, E) be a graph and let S ⊆ V . Then, for any
tree decomposition of G there is at least one vertex v ∈ S such that there is
a bag that contains N [v] ∩ S.
Proof. Let T = (X1 ∪ . . . ∪ Xr , E) be a tree decomposition of G. Fix an
arbitrary vertex u ∈ S and denote by X u any bag in T that contains u.
101
Now, choose a vertex v ∈ S such that the length of the path from X u to
the first bag that contains v is maximum. Denote this bag by X v . We
show that N (v) ∩ S ⊆ X v . Suppose that there is a neighbor w ∈ N (v) ∩ S
that is not contained in X v . Since w and v are adjacent they are together
contained in at least one bag. Since X v is the first bag containing v on
the path from X v to X u and the bags containing w induce a connected
component, from w ∈
/ X v and w ∈ N (v) it follows that the path from X u to
the first bag containing w is via X v and thus longer; a contradiction to the
choice of v.
ω
Theorem 4.3.6. 2-Club is solvable in 2O(2
treewidth the input graph.
)
· n time where ω denotes the
Proof. Let (G, k) be an input instance of 2-Club and let (X1 ∪ . . . ∪ Xr , E)
be a nice tree decomposition for G of width ω. Fix a maximum-size 2-club S.
By Lemma 4.3.5 there is a vertex v ∈ S such that N [v] ∩ S ⊆ X for a bag X.
Let Nv = N [v] ∩ S ⊆ X. First, since r = O(n) there are O(n · ω · 2ω ) cases
for choosing the bag X, the vertex v and its neighbors Nv in X. After having
done this, we root the nice tree decomposition in X. Furthermore, we may
assume that X = Nv , as otherwise one can add a path of forget nodes that
starts in X and step-wisely deletes all vertices in X \ Nv . Next, we describe
a bottom-up dynamic programming algorithm.
Denote by 2P the set of all subsets of a set P . Let Xi be an arbitrary bag.
We have for each combination of some P ⊆ Xi and some T ⊆ 2P an integer
table entry Tabi (P, T ) which is the size of the largest set Ki (P, T ) fulfilling
the following properties:
1. Ki (P, T ) is a subset of the vertices in the subtree rooted in Xi and
Ki (P, T ) ∩ Xi = P ,
2. each type T ⊆ P is in T if and only if there is a vertex v ∈ Ki (P, T ) \ P
of type T , that is, N (v) ∩ P = T , and
3. in G[Ki (P, T )] each vertex in Ki (P, T ) has distance at most two to all
vertices in Ki (P, T ) \ P .
Intuitively, Ki (P, T ) is almost a 2-club as only the vertices from P are
allowed to have distance more than two and all vertices not in Xi are of one
ω
of the types in T . Clearly, the table has at most 2ω · 22 entries per bag and
2ω
we show how to compute each entry in O(2 ) time. This implies, together
ω
with first step to guess X and Nv , the claimed running time of 2O(2 ) · n.
Before showing how to compute the table entries, we prove the claim that
a largest value Tab(Nv , T ) for any T in the table of the root X is equal to
the size of a largest 2-club in G that contains Nv :
First, consider a table entry Tab(Nv , T ). By Property 1 it follows
that Nv = K(Nv , T ) ∩ X and since Nv ⊆ N [v] it follows by Property 3
102
that K(Nv , T ) is a 2-club. Hence, |S| ≥ |K(Nv , T )|. In the other direction,
let S 0 be a 2-club in G with Nv ⊆ S 0 . Let T be the set containing all T ⊆ Nv
where a vertex u ∈ S 0 \ Nv with N (u) ∩ Nv = T exists. By definition, S 0 fulfills Properties 1 & 2 for K(Nv , T ) and since S 0 is a 2-club also Property 3,
implying that |S 0 | ≤ K(Nv , T ).
Computation of the table entries. We now specify how to compute the
table Tabi (P, T ) for a bag Xi by distinguishing whether Xi is a leaf, an introduce node, a join node, or a forget node. Table entries where a corresponding
set fulfilling Properties 1 to 3 does not exist are set to -∞.
Leaf node: Let Xi be a leaf in the tree decomposition. Clearly, since
K(P, T ) has to be a subset of vertices in the subtree of Xi (Property 1), by
Property 2 we only have to consider the case where T = ∅. Then K(P, T )
is equal to P , hence, we set Tabi (P, T ) = |P | if T = ∅ and otherwise
Tabi (P, T ) = −∞.
Introduce node: Let Xi be an introduce node with the child node X` and
let u ∈ Xi \ X` be the introduced vertex. First, assume u ∈
/ P . Then, by
Property 1 it is correct to set Tabi (P, T ) to the value of Tab` (P, T ).
Second, assume that u ∈ P . By Property 1 all vertices in Ki (P, T ) \ P are
not in Xi and since u ∈ Xi \ X` , if there is a type T ∈ T with u ∈ T , then
we set Tabi (P, T ) = −∞ as Property 2 cannot be fulfilled. Additionally,
if there is a type T ∈ T with T ∩ N (u) = ∅, then also set Tabi (P, T ) =
−∞ as Property 3 cannot be fulfilled. In all other cases, observe that
K` (P \ {u}, T ) ∪ {u} fulfills all properties for Ki (P, T ), hence, it is correct
to set Tabi (P, T ) = Tab` (P \ {u}, T ) + 1.
Forget node: Let Xi be a forget node with the child node X` and u ∈
/ P and because of Properties 1 & 2, u ∈ K(P, T ) can
X` \ Xi . Since u ∈
be true only if Tu = N (u) ∩ P ∈ T . Hence, if Tu ∈
/ T , then set Tabi (P, T )
equal to Tab` (P, T ).
Consider the remaining case where Tu ∈ T . Let Kiu (P, T ) be a maximum
set fulfilling Properties 1 to 3 for Ki (P, T ) such that u ∈ Kiu (P, T ). Then,
Kiu (P, T ) either contains only vertex u as Tu -type vertex or at least two
vertices of type Tu . If there are at least two, then |Kiu (P, T )| = |K` (P ∪
{u}, T )|. In case of one Tu -type vertex, the size of Kiu can be at most
equal to those of K` (P ∪ {u}, T \ {Tu }). However, to ensure Property 3 in
this case one has to additionally check whether vertex u has in G[K` (P ∪
{u}, T \ {Tu })] distance at most two to all vertices in P (by Property 2
this can be checked by just knowing P and T ). If this check was positive,
then we have |Kiu (P, T )| = max{|K` (P ∪ {u}, T )|, |K` (P ∪ {u}, T \ {Tu })|},
otherwise |Kiu (P, T )| = |K` (P ∪ {u}, T )|.
Finally, since either u ∈ Ki (P, T ) or not, it follows in case of Tu ∈ T that
Tabi (P, T ) = |Ki (P, T )| = max{|Kiu (P, T )|, Tab` (P, T )}.
103
Join node: Let Xi be a join node and let X` and Xj be the two child
nodes with Xi = X` = Xj . We call two subsets T` , Tj ⊆ T consistent if
T` ∪ Tj = T and for any two types T, T 0 ∈ T with T ∩ T 0 = ∅ it either holds
that T, T 0 ∈ T` or T, T 0 ∈ Tj . We prove that it is correct to set
Tabi (P, T ) =
max
∀ consistent T` ,Tj ⊆T
Tab` (P, T` ) + Tabj (P, Tj ) − |P |.
Let Ki` (Kij ) be the intersection of Ki (P, T ) with the vertices in the subtree
rooted in X` (Xj , respectively). Clearly, Ki` ∩ Kij = P . Additionally, let
T` ⊆ T (Tj ⊆ T ) be the types of the vertices in Ki` (Kij , respectively).
Observe that, by Property 3 each vertex u ∈ Ki` \ P has distance at most
two to any v ∈ Kij \ P . However, by the properties of a tree decomposition
u and v cannot be adjacent and thus N (u) ∩ N (v) ⊆ P . Hence the types
of u and v have a non-empty intersection, implying that T` and Tj are
consistent. Moreover, Ki` fulfills all properties for K` (P, T` ) and Kij fulfills
all properties for Kj (P, Tj ). Note that this is because removing from Ki (P, T )
a set of vertices A with either A ⊆ Ki` \ P or A ⊆ Kij \ P may violate only
Property 2. Hence, there are consistent T` and Tj such that |Ki (P, T )| ≤
|K` (P, T` )| + |Kj (P, Tj )| − |P |.
In the other direction, one can see that for each pair of consistent sets
T` , Tj ⊆ T it holds that K` (P, T` ) ∪ Kj (P, Tj ) fulfills all properties (except
maximality) for Ki (P, T ): Since Xi = X` = Xj it is clear that Property 1 is
fulfilled. Moreover, as for each type T ∈ T it holds that T ∈ T` or T ∈ Tj ,
also Property 2 is fulfilled. Finally, by the definition of consistency, also
Property 3 is fulfilled.
4.4 Kernelization: Algorithms and Lower
Bounds
In this section, we provide polynomial kernels for 2-Club parameterized by
a cluster editing set and a feedback edge set, respectively. While these
parameters can often be rather large, we show that for the (also relatively large) parameter vertex cover, there exists no polynomial kernel
(unless NP ⊆ coNP/poly). Indeed, they are the only parameters in our
structural parameter hierarchy (Figure 4.2) that admit a polynomial kernel.
4.4.1 A Quadratic-Vertex Kernel for Cluster Editing Set
Size
We show how to obtain an O(k 2 )-vertex kernel when k is the size of a (not necessarily of minimum cardinality) cluster editing set. The computational task
104
to compute a minimum-size cluster editing set is known as the NP-complete
Cluster Editing problem. The parameterized complexity of Cluster
Editing has been extensively studied, for a survey see Böcker and Baumbach [BB13], and there is also a polynomial-time 2.5-factor approximation
algorithm for it [vZW09].
Let G = (V, E), an integer `, and a cluster editing set D be an instance of
2-Club; the parameter is k = |D|. Denote by V (D) the set of all endpoints
of the edges in D and observe that G − V (D) is a cluster graph, that is, all
connected components of G − V (D) are cliques. The following two rules
yield an O(k 2 )-vertex kernel for 2-Club.
Reduction Rule 4.4.1. Let C be a cluster in G − V (D) and set DC =
NG (v) ∩ V (D) for some v ∈ C. If C or NG [C ∪ DC ] is a 2-club of size at
least `, then reduce to a constant-size yes-instance. Otherwise, if |DC | ≤ 1,
then remove C.
Lemma 4.4.2. Reduction Rule 4.4.1 is correct and can be exhaustively
applied in O(n2 m) time. Furthermore, the resulting graph G0 has at most k
clusters in G0 − V (D) and each of them has size less than `.
Proof. We first prove correctness. Observe that NG (v) = NG (w) for any two
vertices v and w in a cluster C of G − V (D). Clearly, if C or NG [C ∪ DC ]
are 2-clubs of size at least `, it is correct to reduce to a yes-instance. Hence,
in case DC = ∅ it is correct to delete C, because C is an isolated clique
with |C| < `. In the remaining case let |DC | = 1, implying that C ∪ DC
is a clique. Thus the set NG [C ∪ DC ] is 2-club and it is the largest 2-club
containing any vertex from C.
After applying Reduction Rule 4.4.1 it holds that |DC | > 1 for each
cluster C in G0 − V (D) and since |V (D)| ≤ 2k this implies that the number
of clusters is at most k and each of them has size less than `. For the
running time, clusters in G − V (D) can be identified in O(n + m) time and
testing whether a vertex subset is a 2-club can be done in O(nm) time via a
breath-first search starting in each vertex.
In the following assume that G is reduced with respect to Reduction
Rule 4.4.1. Since each cluster in G−V (D) has size at most `−1 (Lemma 4.4.2)
it follows that if ` ≤ 2k + 1, then there are at most 2k 2 + 2k vertices left and
we are done.
Now, consider the case where ` > 2k + 1. To bound the size of the clusters
in G − V (D) we use the following observation. Its correctness follows from
the fact that two vertices in different clusters of G − V (D) are not adjacent
and have no common neighbor.
105
Observation 4.4.3. For every 2-club S in G there is at most one cluster C
in G − V (D) such that S has a nonempty intersection with C.
Observation 4.4.3 implies that every 2-club of size at least ` contains at least
` − 2k vertices from exactly one cluster C of G−V (D). Since all vertices in C
are twins, Observation 4.1.1 now implies that an inclusion-maximal 2-club
either contains all or no vertices from C. Hence, for ` > 2k + 1 decreasing `
and the size of each cluster C by ` − 2k − 1 produces an equivalent instance.
This leads to the following data reduction rule.
Reduction Rule 4.4.4. Delete `−2k−1 arbitrary vertices in each cluster C
of G − V (D) and set ` = 2k + 1.
Note that if |C| ≤ l − 2k − 1, then we simply delete all vertices of C. After
an exhaustive application of Reduction Rule 4.4.4 for each cluster C it holds
that 1 ≤ |C| < 2k + 1. Thus, we arrive at the following.
Theorem 4.4.5. 2-Club parameterized by the cluster editing set size k
admits an (2k 2 + 2k)-vertex kernel that can be computed in O(n2 m) time.
Proof. Let I = (G, `, D) be an instance where Reduction Rules 4.4.1 & 4.4.4
have been exhaustively applied. Since I is reduced with respect to Reduction
Rule 4.4.1, there are at most k clusters each of size less than ` in G − V (G).
Finally, ` ≤ 2k + 1 due to Reduction Rule 4.4.4. Altogether this implies
that G contains at most 2k 2 + 2k vertices.
As to the running time, Reduction Rule 4.4.1 can be performed in O(n2 m)
time (Lemma 4.4.2). The clusters in G − V (D) can be computed in O(n + m)
time and deleting ` − 2k − 1 vertices in C (Reduction Rule 4.4.4) is clearly
doable in O(|C|) time. Hence, the corresponding running time is dominated
by the application of Reduction Rule 4.4.1.
The correctness of Reduction Rule 4.4.4 is based on Observation 4.4.3
which is only valid for 2-clubs. For s > 2 it is open whether there exists a
polynomial kernel for s-Club parameterized by a cluster editing set. However,
the more general case of Weighted s-Club, where the vertices
P have positive
weights ω and the task is to find an s-club S such that ω(S) = v∈S ω(v) ≥ `,
has even a linear vertex kernel with respect to cluster editing. It is motivated
by the fact that all vertices in a cluster C of G − V (D) are twins and
thus by Observation 4.1.1 either all or none of them are contained in a
maximum s-club.
The linear vertex kernel for Weighted s-Club is an adaption of the
quadratic vertex kernel for 2-Club (see Theorem 4.4.5). We first adapt
Reduction Rule 4.4.1 for weights. To this end, note that the proof of
Lemma 4.4.2 uses the restrictions of 2-clubs only if N [C ∪ DC ] is a 2-club.
106
Reduction Rule 4.4.6. If there is a cluster C in G − V (D) with ω(C) ≥ `,
then reduce to a constant-size yes-instance. Otherwise, if N (C) ∩ V (D) = ∅,
then delete C.
After exhaustively applying Reduction Rule 4.4.6, there are at most 2k
remaining clusters in G as each cluster has at least one neighbor in V (G) and
|V (G)| ≤ 2k. The next data reduction rule uses the weights on the vertices
in order to merge the remaining clusters in G − V (D) into one vertex.
Reduction Rule 4.4.7. For each cluster C in G − V (D), delete all but one
vertex in C and set the weight of the remaining vertex to ω(C).
The correctness of the data reduction rule follows from Observation 4.1.1.
After applying Reduction Rule 4.4.7, each cluster C has size one and, hence,
the application of Reduction Rules 4.4.6 & 4.4.7 leads to the following.
Theorem 4.4.8. Weighted s-Club parameterized by the size k of a cluster
editing set admits an O(n + m)-time computable 4k-vertex kernel.
As a consequence of Theorem 4.4.8, there is a single-exponential time
algorithm solving (weighted) s-Club parameterized by a cluster editing set.
Corollary 4.4.9. Weighted s-Club for s ≥ 2 parameterized by the size k
of a cluster editing set can be solved in O(6.88k · k 3 + n + m) time.
Proof. First, apply Reduction Rules 4.4.6 & 4.4.7 and thus by Theorem 4.4.8
the remaining instances consists of at most 4k vertices. Hence, applying
the search tree algorithm running in O(1.62n · nm) time [Cha+13] yields an
O(6.88k · k 3 + n + m)-time algorithm.
4.4.2 A Linear Kernel for Feedback Edge Set Size
Let (G, `) be a 2-Club-instance and let F ⊂ E be a feedback edge set of
G = (V, E). In the following, we present three reduction rules for 2-Club
and show that exhaustively applying these rules yields a kernel with at
most 5 · |F | vertices and, thus, at most 6 · |F | − 1 edges. The correctness of
the first data reduction rule follows from the fact that for each vertex v the
set N [v] is a 2-club.
Reduction Rule 4.4.10. If there is a vertex v ∈ V with |N [v]| ≥ `, then
reduce to a constant-size yes-instance.
In the following, we exploit that after application of Reduction Rule 4.4.10
all 2-clubs of size at least ` have to “use” feedback edges. The next rule
removes all vertices that are not on paths between the endpoints of feedback
edges. These vertices are defined as follows. Let T = (V, E \ F ) be the forest
obtained by deleting F from G.
107
Definition 4.4.11 (Feedback Edge Path). For a feedback edge {u, v} ∈ F
the path P{u,v} between u and v in T is called feedback edge path. If a
vertex w lies on the path P{u,v} , then the edge {u, v} is a spanning feedback
edge of w.
Reduction Rule 4.4.12. Let (G, `) be reduced with respect to Reduction
Rule 4.4.10. Then, delete all vertices that do not lie on any feedback
edge path.
Proof of correctness for Reduction Rule 4.4.12: Let v be a vertex that does
not lie on any feedback edge path. Then, v is not contained in a cycle in G.
Suppose otherwise, and let C be a cycle that contains v. Clearly, C contains
at least one feedback edge. In case C contains ` > 1 feedback edges, then a
cycle C 0 with ` − 1 feedback edges also containing v is obtained by replacing
an arbitrary feedback edge with a path consisting only of edges from E \ F
(by the minimality of F such a path exist). In case C contains exactly one
feedback edge, v lies on the feedback edge path of this edge, contradicting
the initial assumption for v.
Let S be a 2-club containing v. Since v is not contained in any cycle in G,
v has degree one in G[S] or deleting v disconnects G[S]. In the first case, S
is completely contained in the neighborhood of v’s neighbor in G[S]. In
the second case, S ⊆ N [v]. Since G is reduced with respect to Reduction
Rule 4.4.10, S has size less than `. Consequently, there is no 2-club of
size at least ` that contains v. Therefore, removing v from G yields an
equivalent instance.
The final rule removes vertices that are too far away from feedback edges.
Reduction Rule 4.4.13. If there is a vertex v that has in G distance at
least three to at least one endpoint of every spanning feedback edge, then
remove v.
Proof of correctness for Reduction Rule 4.4.13: We show that v is not contained in a 2-club of size at least `. Let S be a 2-club containing v. Since v
has in G distance at least three to at least one endpoint of every spanning
feedback edge, S contains at most one endpoint of every spanning feedback
edge. Consequently, either v has degree one in G[S] or deleting v disconnects G[S]. In the first case, S is completely contained in the neighborhood
of v’s neighbor in G[S]. In the second case, S ⊆ N [v]. Since G is reduced with
respect to Reduction Rule 4.4.10, S has size less than `. Consequently, there
is no 2-club of size ` that contains v. Therefore, removing v from G yields
an equivalent instance.
Exhaustively applying these data reduction rules results in a linear kernel:
108
u
x
y
z
v
w
Figure 4.4: Illustration of the definitions in the proof of Theorem 4.4.14. Herein,
bold edges are feedback edges, regular edges are edges of T . The path P{u,v} connects the endpoints of the feedback edge {u, v} in T . The feedback edge {u, w} covers the feedback edge {u, v}; the feedback edge {u, v} is u-minimal; the u-neighbor
of {u, v} is x, the v-neighbor is z; the vertices x, y and z are all satisfied.
Theorem 4.4.14. The 2-Club problem parameterized by the size k of a feedback edge set admits a kernel of size 6k that can be computed in O(n4 ) time.
Proof. Let G be a graph being reduced with respect to Reduction Rules 4.4.10,
4.4.12 and 4.4.13. The size of G can be bounded as follows. There are at
most 2k vertices incident with feedback edges; let X denote this vertex set.
To show the kernel size, we show that the set Y = V \ X of vertices not
incident with any edge in F has size at most 3k. The main idea underlying
our proof is that after the reduction rules have been exhaustively applied,
each vertex of Y is in T either adjacent to an endpoint of some feedback
edge or has distance exactly two to both endpoints of a spanning feedback
edge. Using this characterization, a size bound of 5k for Y is relatively easy
to achieve. In the following, we use a more sophisticated approach to show
that |Y | ≤ 3k.
The main idea of the proof is to iteratively add feedback edges to T , and
count the number of vertices for which there is a spanning feedback edge
whose endpoints have distance at most two to this vertex. More precisely,
let T0 = T and let Tk = G. Furthermore, let Ti be obtained from Ti−1 by
adding a feedback edge e with the following property: all feedback edges e0
such that Pe is a subpath of Pe0 are already contained in Ti−1 ; in the following,
we say that Pe0 covers Pe (see Figure 4.4 for an illustration). Note that
finding such an edge e is always possible: If e is covered by an edge e0 that
is not contained in Ti−1 , then we consider e0 . Clearly the path Pe0 is longer
than the path Pe . Therefore, there has to be some edge whose covering edges
are already contained in Ti−1 .
For each Ti we bound the number of vertices that have distance at most
two to both endpoints of a spanning feedback edge; we call these vertices
satisfied. To this end, we create a set Yi ⊆ {(v, e) | v ∈ V, e ∈ F }. This set
contains pairs of satisfied vertices and spanning feedback edges to which
they are attributed. The central property for Yi is that for each satisfied
vertex in Ti the set Yi contains at least one pair that contains this vertex.
109
Then, the aim is to show that |Yk | ≤ 3k.
Before proving the claim we introduce some further terminology concerning
the feedback edges that will be used in the proof. A feedback edge {u, v} is
called u-minimal in Ti if there is in Ti no feedback edge {u, w} such that the
path P{u,w} is a subpath of P{u,v} . A vertex w is a u-neighbor of a feedback
edge {u, v} if {u, v} spans w and w is in T a neighbor of u.
Now, by induction on i we show that for each i there is a set Yi with the
following properties:
• |Yi | ≤ 3i,
• each vertex satisfied in Ti is in at least one pair contained in Yi , and
• if a feedback edge {u, v} is u-minimal in Ti , then Yi contains the
pair (w, {u, v}) where w is the u-neighbor of {u, v}.
The above properties can be proven as follows. By definition, T1 contains
one feedback edge {u, v}. For each satisfied vertex w we add the pair
containing (w, {u, v}) to Y1 . Then Y1 fulfills the invariant since at most three
vertices are satisfied in T1 : the u-neighbor of {u, v}, the v-neighbor of {u, v}
and at most one further inner vertex (this is the case when P{u,v} contains
exactly five vertices).
For the inductive step, assume that the claim holds for i − 1, and let {u, v}
be the feedback edge that is added from Ti−1 to Ti . We construct Yi fulfilling
the claim as follows. Initially, we set Yi = Yi−1 . Then, for the u-neighbor
and the v-neighbor of {u, v}, we add the pair containing {u, v} and the
respective vertex to Yi , and, if such a vertex exists, a pair containing the
vertex that is in T adjacent to both inner neighbors of {u, v}. After these
additions, |Yi | ≤ |Yi−1 | + 3. We now show that for each further satisfied
vertex that is not satisfied in Ti−1 we can add a pair containing this vertex
while also removing another pair from Yi without violating the invariant.
All further vertices that are satisfied in Ti but not in Ti−1 are neighbors
of u or v since they must “use” {u, v} in order to have distance two to
some endpoint of a spanning edge. Since the neighbors of u and v that
are in P{u,v} have already been added to Yi , only satisfied vertices that are
not in P{u,v} have to be considered. Let Z denote this set and partition Z
into Zu = Z ∩NT [u] and Zv = Z ∩NT [v]. Clearly, such a partition is possible
since every vertex of Z is in T a neighbor of either u or v but not of both.
Now consider the set Zv , and let z1 , . . . , zq denote the vertices in Zv .
By definition, each zj is adjacent to v and there is be a spanning feedback
edge e such that in Ti but not in Ti−1 the vertex zj has distance at most
two to both endpoints of e. Consequently, u is one of the endpoints of e,
that is, e = {u, wj } for some wj . Moreover, {u, wj } covers {u, v} and,
110
u
x
v
z1
w1
z2
w2
...
zq
wq
Figure 4.5: Illustration of the vertices in Zv and of the corresponding spanning
edges.
furthermore, for all zj ’s the wj ’s are pairwise distinct. Thus, the situation
is as depicted in Figure 4.5. We now show that for each zj there is a
pair (x, {u, pj }) where x is the u-neighbor of {u, v} in Yi that can be removed
from Yi without violating the invariant. By the invariant, there is for each
spanning edge {u, wj } a u-minimal edge ej that is covered by {u, wj } and
such that Yi−1 contains the pair (x, ej ). Since Yi is a superset of Yi−1
this pair is also contained in Yi . Note that since the last edge that was
added is {u, v} and by the fact that Ti contains no edges that are covered
by {u, v} these pairs are pairwise distinct: they must cover {u, v} and thus
their endpoints pj are on the paths from v to wj ; these paths are, with the
exception of v, vertex-disjoint. Now each such pair can be removed, since
after adding {u, v} each edge {u, pj } is not u-minimal, and we have already
added the pair (x, {u, v}). Consequently, when adding the |Zv | pairs for the
satisfied vertices in Zv to Yi we can at the same time safely remove |Zv |
pairs without violating the invariant.
Analogously, when adding the pairs for Zu , we can also remove |Zu | pairs
without violating the invariant. Summarizing, we have
|Yi | ≤ |Yi−1 | + 3 + |Zv | − |Zv | + |Zu | − |Zu | = 3 · (i − 1) + 3 = 3i.
The overall kernel bound now follows from |Y | ≤ |Zk | ≤ 3k.
We complete the proof by bounding the running time. Clearly, Reduction
Rule 4.4.10 can be exhaustively applied in linear time. The applicability
of Reduction Rule 4.4.12 can be checked in O(nm) time by considering
each edge in F and labeling the vertices on its feedback edge path. After
111
computing an all-pairs shortest path matrix in O(n3 ) time, the applicability
of Reduction Rule 4.4.13 can be checked in O(nm) time. The overall running
time now follows from the fact that Reduction Rule 4.4.12 and Reduction
Rule 4.4.13 can be applied at most n times.
4.4.3 Lower Bounds with Respect to Bandwidth and
Vertex Cover
We show that 2-Club does not admit a polynomial kernel with respect
to the parameters vertex cover and bandwidth. The first result implies
that 2-Club does not admit a polynomial kernel for many structural graph
parameters such as feedback vertex set, maximum degree, or treewidth (see
the block “rooted” in parameter vertex cover in Figure 4.2).
We start with proving the kernelization lower bound for bandwidth and
we observe that a linear-vertex Turing kernel can be achieved.
Proposition 4.4.15. 2-Club parameterized by the bandwidth k does not
admit a polynomial kernel unless NP ⊆ coNP/poly; it admits a 2k-vertex
Turing kernel which can be computed in O(nm) time.
Proof. Having a set of input instances of 2-Club, taking the disjoint union
of the corresponding graphs gives a simple OR-cross-composition 2-Club
(Definition 2.3.10) and proves that 2-Club does not admit a polynomial
kernel with respect to the bandwidth unless NP ⊆ coNP/poly.
We next show that 2-Club parameterized by bandwidth has a linear
Turing kernel. Let G = (V, E) be a graph with bandwidth k and let the
vertices be labeled v1 , . . . , vn such that for each edge {vi , vj } ∈ E it holds
that |i−j| ≤ k. The Turing kernel can be obtained by creating for each vi ∈ V
the graph G[{vi , vi+1 , . . . , vi+2k }]. The idea of the Turing kernel is simply to
try all possibilities to choose the vertex vi which has the lowest index among
all 2-club vertices. By definition of bandwidth, all vertices within distance
two of vi are in {vi+1 , vi+2 , . . . , vi+2k } and, hence, S ⊆ {vi , vi+1 , . . . , vi+2k }.
Thus, the Turing kernel follows.
The hardness result in Proposition 4.4.15 directly implies that 2-Club also
has no polynomial kernel for the parameter maximum degree. Next, we show
that 2-Club parameterized by vertex cover does not admit a polynomial
kernel. This implies the same lower bound for many other weaker parameters
such as a feedback vertex set, treewidth, degeneracy, distance to cluster
graphs, etc. (see Figure 4.2).
Theorem 4.4.16. 2-Club parameterized by a vertex cover has no polynomial kernel unless NP ⊆ coNP/poly.
112
v21 , . . . , v2n
v31 , . . . , v3n
v51 , . . . , v5n
v1
v2
e2,3
e2,3
v3
e3,5
VE
e3,5
v4
v5
G
v1
v4
G0
Figure 4.6: Example of the construction in the proof of Theorem 4.4.16. The
graph G is an input for Clique parameterized by vertex cover. The gray vertices
are a vertex cover in G. The graph G0 is constructed as described in the proof of
Theorem 4.4.16.
Proof. We give a polynomial-time and parameter reduction (Definition 2.3.12)
from Clique parameterized by vertex cover to 2-Club parameterized by
vertex cover. Unless NP ⊆ coNP/poly, Clique does admit a polynomial
kernel with respect to vertex cover [BJK11].
Let (G = (V, E), X, k) be an instance of Clique parameterized by a
vertex cover X. We construct a graph G0 as follows (for an illustration see
Figure 4.6). First, add for each vertex vi ∈ X exactly n vertices {vi1 , . . . , vin }
to G0 . The construction will ensure that these n vertices are twins in G0 .
Next, add a set VE of “edge-vertices” as follows. For each edge {vi , vj } ∈ E
between two vertex cover vertices vi , vj ∈ X add an edge-vertex ei,j to G0
and make ei,j adjacent to all vertices in {vi1 , . . . , vin , vj1 , . . . , vjn }. Then, add
edges such that VE is a clique. The idea of the construction is to ensure
that if a 2-club contains two vertices vix and vjy , then vi and vj are adjacent
in G0 . Hence, a 2-club containing such vertices corresponds to a clique in G.
In order to handle the case where a size-k clique K in G contains a
vertex from V \ X, add the vertex set V \ X to G0 . Then, for each added
vertex v ∈ V \ X and each edge-vertex ei,j ∈ VE add an edge if v is adjacent
to vj and vi in G. Observe that the construction runs in polynomial time,
that it ensures that VE is a vertex cover for G0 , and that |VE | ≤ |X|
2 . Thus,
it is a polynomial-time and parameter reduction. We complete the proof
by showing that G has a clique of size k ⇔ G0 has a 2-club of size at least
|VE | + (k − 1)n + 1.
“⇒”: Let K be a size-k clique in G. Then, every pair of vertices vi , vj ∈
K ∩ X has the common neighbor ei,j in G0 . Hence, by Observation 4.1.1,
the vertex set S containing the twins of all vertices in K ∩ X and VE is
113
a 2-club. In case K ⊆ X, this 2-club is of size |VE | + kn. Otherwise, one
can add the vertex v ∈ K \ X to S. Then, S has size |VE | + (k − 1)n + 1
and it is also a 2-club: each vertex vix ∈ S \ VE has with v the common
neighbor ei,j where vj is some other vertex in K; similarly, each vertex in VE
has a common neighbor with v.
“⇐”: Let S be a 2-club of size at least |VE | + (k − 1)n + 1 in G0 . A twin-free
set in S is a subset of S \ (VE ∪ (V \ X)) that does not contain twins. First,
since |V \ X| < n + 1, it follows that S contains a twin-free set of size at
least k − 1. Moreover, for each vertex pair {vi , vj } in a twin-free set, the
vertex ei,j has to be contained in S as well: otherwise vi and vj have distance
greater than two in G[S]. Thus, vi is adjacent to vj in G. Therefore, a
twin-free set in S corresponds to a clique in G. Hence, if there is a twin-free
set of size at least k, then there is a size-k clique in G. It thus remains to
consider the case where a largest twin-free set T is of size k − 1. In this
case, S contains at least one vertex v ∈ V \ X. Since S is a 2-club, there
is for each vertex vi ∈ T at least one edge-vertex ei,j ∈ S that is adjacent
to vi and v. By construction, this implies that vi is adjacent to u in G.
Consequently, T ∪ {v} is a size-k clique in G.
4.5 On the Optimality of the Dual Parameter
Algorithm
In this section, we prove running time and kernelization lower bounds for
s-Club when parameterized by the dual parameter k 0 = n − `. We first
show that there is a polynomial-time reduction from Sat to s-Club with
certain properties that allows to infer these lower bounds.
As a side result, answering an open question by Liu et al. [LZZ12] the
reduction will also prove that the s-Club Cluster Vertex Deletion
problem does not admit a polynomial kernel for all s ≥ 2. Therein, the
problem is to decide for a given graph G and an integer k whether by deleting
at most k vertices in G one could get a graph whose connected components
are 2-clubs.
Lemma 4.5.1. For any s ≥ 2 there is a polynomial-time reduction from
Sat to s-Club that computes for any n-variable CNF formula an equivalent
s-Club-instance (G, `) with dual parameter k 0 = n such that there are
exactly k 0 pairwise disjoint vertex pairs each having distance s + 1.
Proof. Let F be a CNF formula with n variables and assume without loss
of generality that F does not contain a clause that contains the positive and
the negative literal of the same variable. We describe how to construct an
114
lvx1 ,vx2
L
b 2s c
vx 1
c1
c2
vx 1
vx 2
V
C1
c1
c2
vx2
vx 3
c1
vx3
c2
C2
C = C1 ∪ C2 ∪ C3
C3
Figure 4.7: Illustration of the construction in the proof of Lemma 4.5.1 for the
CNF formula consisting of the clauses C1 = (x1 ∨ ¬x2 ), C2 = (¬x1 ∨ x3 ), and
C3 = (¬x2 ∨ ¬x3 ). Gray edges indicate a length-b 2s c path between the endpoints
which is explicitly drawn for lvx1 ,vx2 and its two neighbors.
n0 -vertex graph G such that for k 0 = n the graph G has an s-club of size
at least n0 − k 0 if and only if F has a satisfying assignment (see Figure 4.7
for an illustration): First, add the literal vertex set V to G, that is, a set
that contains for each variable x in F the two vertices vx , vx where vx
is the positive literal vertex that corresponds to x and vx is the negative
literal vertex that corresponds to ¬x. Next, add the clause vertex set C
that contains for each clause C in F two clause vertices c1 , c2 . Additionally,
if the literal x (¬x) occurs in the clause C, then make the corresponding
clause vertices c1 and c2 adjacent to the positive (negative) literal vertex vx
(vx , respectively).
The basic idea of the construction is that for each variable x the two
vertices vx and vx have distance s + 1 and thus one has to delete one of
them. These are the k 0 = n vertex pairs corresponding to the requirements
on G in Lemma 4.5.1. The clause vertices c1 , c2 for a clause C have the
literal vertices corresponding to the literals in C as common neighbors and
there is no path of length at most s between them that avoids all of these
common neighbors. Hence, the literal vertices remaining after deleting for
each variable either the corresponding positive literal or negative literal
vertex correspond to a satisfying assignment for F if and only if for each
clause C the clause vertices c1 , c2 still have at least one common neighbor
(which is a literal vertex).
115
We extend the graph G to fulfill to above mentioned properties: For each
vertex pair {u, w} ⊆ V ∪ C not equal to {vx , vx } for same variable x and also
not equal to {c1 , c2 } for some clause C, add a vertex lu,w , add a length-b 2s c
path from lu,w to w, and also a length-b 2s c path from lu,w to u. This implies
that dist(u, w) = s in case of even s and dist(u, w) = s − 1 otherwise. Collect
all vertices lu,w to the set L and denote by P all vertices on the b 2s c-length
paths except the endpoints. Clearly, in case of s ∈ {2, 3} all the paths
inserted above are only edges between the endpoints and thus P = ∅.
Finally, if s is odd, then add a distinguished vertex l which is adjacent to
each vertex in L and otherwise, if s is even, add edges such that L is a clique.
This ensures that a pair {u, w} ⊆ V ∪C has distance at most s if and only if
there is some vertex lu,w ∈ L or {u, w} is equal to the clause vertices {c1 , c2 }
for some clause C. We next prove that (G, `) with ` = n0 −k 0 is a yes-instance
of s-Club ⇔ F has a satisfying assignment.
“⇒”: Let D ⊆ V be a set such that |D| ≤ k 0 and G − D is an s-club.
Since the positive vx and negative literal vertex vx of each variable x have
distance s + 1, either vx ∈ D or vx ∈ D and since there are k 0 = n variables
it follows that D ⊆ V. Hence all clause vertices are contained in G − D
and thus the pair {c1 , c2 } for each clause C has distance at most s. This
implies that there is a vertex u ∈ V such that c1 and c2 are adjacent to u.
By the construction, u corresponds to a literal in the clause C and thus the
remaining literal vertices in G−D correspond to a satisfying assignment of F.
“⇐”: Assume that there is a satisfying assignment for F and let D ⊆ V be
the set of literal vertices whose corresponding literals in F are assigned to be
false. Clearly, |D| = k 0 = n and it remains to prove that G − D is an s-club:
By construction, for any two vertices {u, w} ⊆ V ∪ C there is either a
vertex lu,w ∈ L or if {u, w} corresponds to some {c1 , c2 } for some clause C,
then there is common literal neighbor in V not contained in D, implying
that in each case distG−D (u, w) ≤ s. To prove the remaining cases observe
that each vertex in v ∈ V ∪ C ∪ P ∪ L ∪ {l} has a length at most b 2s c path to
at least one vertex in L. Moreover, it holds that the distance from a vertex
in P ∪ L ∪ {l} to any vertex in L is at most b 2s c + 1 in case of odd s and at
most 2s for even s, implying that G − D is a s-club.
Since Lemma 4.5.1 provides a linear-parameter transformation (Definition 2.3.12) from Sat parameterized by the number of variables to 2-Club
parameterized by the dual parameter k 0 , it follows that an algorithm for
0
s-Club running in O((2 − )k · |G|O(1) ) time for some > 0 would disprove
the SETH (Hypothesis 2.3.17). This bound is tight since s-Club can be
0
solved in O(2k · nm) time [Sch+12].
116
Corollary 4.5.2. Unless the SETH fails, s-Club on a graph G parameter0
ized by the dual parameter k 0 = n − ` cannot be solved in O((2 − )k · |G|O(1) )
time for all s ≥ 2.
Dell and van Melkebeek [DvM10] and Fortnow and Santhanam [FS11]
showed that Sat does not admit a polynomial kernel with respect to the
number of variables unless NP ⊆ coNP/poly. Hence, Lemma 4.5.1 also
implies the following lower bound.
Corollary 4.5.3. s-Club for all s ≥ 2 parameterized by the dual parameter k 0 does not admit a polynomial kernel unless NP ⊆ coNP/poly.
Finally, as Lemma 4.5.1 states that the graph constructed in the reduction
from Sat to s-Club has k 0 pairwise disjoint vertex pairs where from each pair
one has to delete at least one vertex, in this special case s-Club Cluster
Vertex Deletion is equivalent to s-Club parameterized by the dual
parameter k 0 . Thus by the same argumentation as for Corollary 4.5.3 the
following holds.
Corollary 4.5.4. s-Club Cluster Vertex Deletion for all s ≥ 2 does
not admit a polynomial kernel unless NP ⊆ coNP/poly.
4.6 Implementation and Experiments
In this section we present our experimental findings for 2-Club. We implemented a search tree algorithm, multiple data reduction rules, and a
Turing kernelization and we combined them in several ways to get multiple exact solving algorithms. We then tested them on randomly created
instances as well as on a collection of real-world instances taken from the
2nd & 10th DIMACS challenge [DIM12, DIM93]. We compare our findings to
the performance of the Gurobi 5.1 [Gur 5.1] solver running the integer linear
programming formulation of Bourjolly et al. [BLP02] and an implementation
of Chang et al. [Cha+13] of the same basic search tree algorithm5 . In the following we first describe the algorithms and their variants, the instances and
the benchmark setting, and, finally, we describe our experimental findings.
4.6.1 Implemented Algorithms
Search Tree-Based Algorithm. We implemented the following search tree
algorithm, briefly denoted by ST, to find a maximum 2-club S in a given
5 The
source code of the C++ program of Chang et al. [Cha+13] is publicly available
(see their paper for the reference, accessed March 2013) and we compiled it using their
MAKE file with the gcc compiler version 4.7.2 (Debian 4.7.2-5).
117
graph G = (V, E): If G is not a 2-club, then find a vertex v ∈ V such
that |N2 (v)| (number of vertices within distance two) is minimum among all
vertices. Then, branch into the cases to either delete v from G or to mark v
to be contained in S and subsequently delete all vertices in V \ N2 [v]. During
branching we maintain a lower bound, that is, the size `0 of a largest 2-club
found so far; this lower bound is initialized by the maximum degree plus
one. The branching is aborted if the current graph has less than `0 vertices.
After exploring all branches, we output the current lower bound (along with
a 2-club of this size).
The above search tree algorithm was introduced by Bourjolly et al. [BLP02].
It is along with a data structure that maintains the two-neighborhood of all
vertices under vertex deletions the algorithm implemented and evaluated
by Chang et al. [Cha+13]. They also proved by a recursion analysis that
its running time can be upper-bounded by O(αn · nm) where α is the
golden ratio [Cha+13]. Schäfer et al. [Sch+12] showed that for the dual
0
parameter k 0 = n − ` this search tree algorithm runs in O(2k · nm) time (if
branching is aborted if more than k 0 vertices have been removed). Note that
by Corollary 4.5.2, the search tree size measured by k 0 cannot be improved
unless the SETH fails.
Our implementation of the search tree algorithm ST is accelerated in each
branching step by the extensive application of the following data reduction
rules. We describe the rules in descending order of observed effectiveness.
Herein, let G = (V, E) be the graph of the current branching step.
R1 Vertex Cover Rule: Let G0 = (V, E 0 ) be the graph where two vertices
are adjacent if and only if they have distance at least three in G.
Observe that the size of a minimum vertex cover of G0 is a lower
bound on the number of vertex deletions that have to be performed
to transform G into a 2-club. We compute a 2-approximate vertex
cover C for G0 that is disjoint to the marked vertices (as they may not
be deleted). If |V | − d|C|/2e is less than the current lower bound, then
abort this branch.
R2 Cleaning conflicts with marked vertices: If there is a vertex v ∈ V that
has distance at least three to a vertex that is marked to be contained
in the 2-club, then delete v. If v is marked, then abort this branch.
R3 Common neighbors of marked vertices: If there are two non-adjacent
marked vertices with only one common neighbor v, then mark v.
R4 Degree-one vertices: Remove each vertex v that has degree one. If v is
marked, then abort this branch.
The correctness of Rules R1–R3 is obvious. Rule R4 is correct since we
initialized our lower bound by a 2-club formed by a maximum degree vertex
118
and thus a larger 2-club cannot contain degree-one vertices (note that Rule R4
is a special case of Reduction Rule 4.4.12).
Turing Kernelization. We implemented the Turing kernelization which was
introduced by Schäfer et al. [Sch+12]. Therein, the basic observation is that
for any 2-club S in a graph G = (V, E) it holds that S ⊆ N2 [v] for all v ∈ S.
Moreover, after applying Reduction Rule 4.4.10 in advance, |N2 [v]| ≤ `2 . We
say that N2 [v] is the Turing kernel for vertex v.
Later on, running an algorithm for 2-Club together with Turing kernelization means that in any step we choose a vertex v, mark v, and run the
algorithm on the graph induced on N2 [v]. We update the current lower
bound `0 , delete v, delete all vertices u where |N2 [u]| ≤ l0 (they cannot be
contained in any 2-club larger than `0 ), and proceed by choosing the next
Turing kernel.6 We implemented three different strategies to choose the
vertex v among all vertices: i) v has minimum degree (DEG), ii) a feedback
edge set in G[N2 [v]] is of minimum size (FES), and iii) the size of N2 [v] is
minimum (N2).
Note that using Turing kernelization with strategy N2 is indeed equivalent
to what the search tree algorithm does: In one case v is contained in the
maximum 2-club S and thus S ⊆ N2 [v], in the other case v is not contained
and can thus be deleted. This observation explains the effectiveness of the
search tree algorithms on the considered real-world data from social network
analysis. There, the smallest two-neighborhood in the graph is typically
much smaller than the entire vertex set.
4.6.2 Results
Setting and Algorithm Variants. We ran all our experiments on an Intel(R)
Xeon(R) CPU E5-1620 3.60 GHz machine with 64 GB main memory under the
Debian GNU/Linux 7.0 operating system. The program is implemented in
Java and runs under the OpenJDK runtime environment in version 1.7.0_03.
The source code is freely available from http://fpt.akt.tu-berlin.de/
two_club/.
We tested our program on random instances as well as on real-world
data from the 2nd and 10th DIMACS challenge [DIM12, DIM93]. We use
the name scheme SOLVER-TK-STRATEGY to denote the different variants of
the solver, that is, SOLVER is one of {ST,ILP,CST} where ILP refers to the
Gurobi solver running the ILP proposed by Bourjolly et al. [BLP02] and
CST refers to the search tree implementation of Chang et al. [Cha+13].
6 Deleting
all vertices whose two neighborhood is of size at most the current lower bound
can be viewed as a data reduction rule which strengthens R4.
119
Table 4.1: Experimental results on random instances with 0.15 density. For each
combination of a, b, and n we created 100 instances by the random graph generator
proposed by Gendreau et al. [GSS93]. Correspondingly, all other entries namely, m
(# edges), ∆G (maximum degree), ∅G (avg. degree), h-index, 2-club (size of the
largest 2-club), and the time in seconds for the solvers ST and CST are the averages
over all these 100 instances.
a
0.0
b
0.3
n
m ∆G ∅G h-index
150 1672 42.2 22.3
29.3
160 1920 44.8 24.0
31.6
170 2155 47.4 25.4
33.3
0.05
0.25
150 1666 38.7 22.2
160 1913 41.0 23.9
170 2151 43.6 25.3
27.7
29.8
31.5
67.5
78.9
87.4
P
0.1
0.2
150 1679 35.8 22.4
160 1915 38.2 23.9
170 2161 40.1 25.4
26.8
28.6
30.3
57.7
65.5
74
P
2-club
80.3
91.5
99.5
P
ST
0.86
0.95
1.77
CST
1.47
1.75
3.68
3.58
10.4
16.8
31
6.9
17
30.8
64.2
58.2
54.4
157
402
112
85.5
272
834
613
1192
TK indicates that Turing kernelization is used (if not, it is just omitted),
and, correspondingly, STRATEGY states one of the strategies {DEG,FES,N2}
used therein. For example, ST-TK-FES denotes the solver running Turing
kernelization with the strategy FES and uses our implementation of the
search tree algorithm to solve each Turing kernel.
Random Instances. As in previous experimental evaluations [Cha+13,
PB12], we use the random graph generator due to Gendreau et al. [GSS93]
where the density of the resulting graphs is controlled by two parameters,
0 ≤ a ≤ b ≤ 1, and the expected density is (a + b)/2.
Table 4.1 summarizes the performance of ST and those of CST on instances
with density 0.15 (see Table 4.4, for a full list with different densities).
As first observed by Bourjolly et al. [BLP02], density 0.15, a = 0.1, and
b = 0.2 produces the hardest instances. The ST-solver solves instances of
this type for n = 170 within 6.6 min whereas CST is more than two times
slower. Note that Pajouh and Balasundaram [PB12] needed about one hour
on instances with n = 150. We observed that the key point for the good
behavior of our algorithm on these instances is the Vertex Cover Rule that
allows quite frequently to prune the search tree. As can be observed in the
120
Table 4.2: Experimental results on random instances with 0.15 density and n = 130,
comparing ILP-based solvers with ST. For each combination of a and b we created
100 instances by the random graph generator proposed by Gendreau et al. [GSS93].
Correspondingly, all other entries namely, m (# edges), 2-club (size of the largest
2-club), and the time in seconds for the solvers (last five columns) are the averages
over all these 100 instances.
a
b
m 2-club ST ILP ILP- TK-Deg ILP- TK-N2 ILP- TK-FES
0.05 0.25 1264 53.7 2.27 18.9
17.8
17.7
17.7
0.1 0.2 1256 45.3 6.64 90.4
88.4
88.2
88.1
full list of results, combining the ST solver with Turing kernelization does
slightly decrease (about 2 %) its performance but still outperforms other
combinations, e. g. ILP-TK. This effect is in stark contrast to the behavior
on real-world instances, as there Turing kernelization yields a dramatic
performance increase. The reason for that is that the largest 2-club contains
about 50 % of the vertices, hence, the beneficial decrease in the graph size
obtained by Turing kernelization is dominated by the time computing it.
Finally, we would like to emphasize that the ILP-solver, as the only
program, makes use of the eight cores of the processor (all other programs
set up only a single thread). However, even when combining the ILP-solver
with Turing kernelization it is, against our a-priori intuition, significantly
slower than the search tree-based algorithms ST and CST. See Table 4.2 for a
comparison of its performance on instances with n = 130. Since ILP was not
able to solve instances with n > 140 within 5 min we did not perform any
further benchmarks for it on random instances.
Real-World Instances. We considered real-world data from the 2nd & 10th
DIMACS challenge [DIM12, DIM93]. To investigate the usefulness of 2-Club
as natural clique relaxation concept, we ran our algorithm on instances from
the clustering category [DIM12]; a standard benchmark for graph clustering
and community detection algorithms. To test our algorithm on large scale
social network graphs we ran it on graphs from the co-author and citation
category [DIM12]. These graphs were obtained by the co-author relationship
or the citation relation among authors listed in the DBLP and Citeseer
database. Moreover, we performed studies on the instances from the clique
category for the 2nd DIMACS challenge [DIM93]. In addition to the DIMACS
instances, we created a further DBLP coauthor graph, which is the largest
instance in our experiments (dblp_thres_1). Table 4.3 shows the results (see
Table 4.5 for a full list).
121
Table 4.3: Experimental results on instances from the DIMACS implementation
challenges [DIM12, DIM93]. The first column denotes the name of the instances,
n the number of vertices, m the number of edges, ∆G the maximum degree, and
2-club denotes the size of the largest 2-club. The last six columns contain for each
solver the required time in seconds and “#TK” denotes number of Turing kernels
needed to solve.
name
10th DIMACS
clustering
email
hep-th
netscience
PGPgiantcompo
power
10th DIMACS
co-author citation
citationCiteseer
coAuthorsCiteseer
coAuthorsDBLP
graph_thres_01
n
1133
8361
1589
10680
4941
∆G 2-club ST-TR-Deg ST-TR-FES ILP-TR-Deg
time #TK time #TK time #TK
71
50
34
205
19
268495 1318
227320 1372
299067 336
715633 804
72
51
35
206
20
42.8
0.94
0.03
1.14
0.25
1319 79.3
1373 49.4
337
85
805 294
171
0
0
0
0
83.2
0.94
0.03
1.15
0.25
0 78.2
0 49.5
0 85.2
0 275
234
0
0
0
0
70.5
0.93
0.03
1.14
0.24
162
0
0
0
0
0 78.7
0 48.8
0 86.7
0 292
0
0
0
0
Since both, ST and CST, rely on space-consuming data structures to quickly
maintain vertex deletions, they are not suitable for graphs with more than
≈1000 vertices. Hence, we performed all tests with solvers where Turing
kernelization is applied. We observed that, since the average degree in realworld graphs is small, Turing kernelization typically produces small graphs
enabling the corresponding solvers to solve them quickly. More specifically,
in its fastest variant, namely ST-TK-DEG, our search tree algorithm together
with Turing kernelization solves all but one instance from the clustering
category within 10 seconds.7 This is a significant performance increase in
comparison to Pajouh and Balasundaram [PB12] who needed up to 70 min for
these instances. Moreover, although the co-author/citation graphs are quite
large (up to 715,000 vertices), Turing kernelization enabled us to handle them
within roughly 1.5 min (in the preliminary version we needed about 30 min).
Interestingly, Turing kernelization with the DEG-strategy turned out to be
the most effective one. For example, on the email graph from the clustering
7 The
version of the ST-TK-DEG-solver presented in previous work [HKN12] was
slightly faster on these instances. This moderate worsening is due to some
heuristic improvements leading to significant accelerations on instances from the
co-author citation category.
122
category, the DEG-strategy yields a speedup factor of two compared to the
other strategies. We conjecture that, since the minimum degree vertex v
forming the Turing kernel N2 [v] can be marked, this allows the solver to
quickly mark a majority of v’s neighbors as well, because otherwise v would
not have a connection to its rather large two-neighborhood. However, a more
careful analysis of this phenomenon is necessary. Not to our surprise after
the results on random data, combining ILP with Turing kernelization is not
competitive with the combination of ST-TK.
We also performed initial experiments with the combination of CST and
Turing kernelization. However, its performance is roughly similar to the
results for ST-TK because instances formed by the Turing kernels are only
moderately hard and thus they are not suitable to spot the difference between the corresponding solvers. Furthermore, our Turing kernelization is
implemented in Java and thus has to write an input file for each Turing
kernel in order to invoke the C++ program CST. Due to this we did not
perform a further systematic study here.
On a majority of the real-world instances we further observed the unexpected behavior that the largest 2-club is “just” a maximum-degree vertex
together with its neighbors.8 From this, the question arises whether the
resulting community structures are meaningful. In a first step to examine
this, we created from a DBLP coauthor graph subgraphs of the pattern
dblp_thres_i where two authors are related by an edge if they coauthored at
least i papers. We expected that for moderate values of i, say 2 or 3, the
resulting (2-club) communities would have a stronger meaning because there
are no edges between authors that are only loosely related. Unfortunately,
even for values up to i = 6 this seems not to be the case. We think the main
reason for this is the large gap between the maximum-degree vertex (with
degree around 1000) and the average degree (less than 10). Thus, there
seem to be some authors that dominate the overall structure because of their
large number of coauthors. Notably, there are only few of these “dominating”
authors: less than 200 authors have more than 200 coauthors.9
Altogether, our experiments demonstrate that for the hardest randomly
created instances as well as for huge real-world instances our implementation
of the search tree algorithm combined with Turing kernelization significantly
outperforms previous implementations.
8 This
seems to be the reason why “deleting vertices with too small two neighborhood”
is quite successful in Turing kernelization. Indeed, together with the (heuristically
chosen) rule that Turing kernelization is disabled as soon as it lowers to graph size by
less than 50 %, it causes the effect that the number of Turing kernels that need to be
solved by the corresponding solver in {ST,ILP,CST} is zero for most instances.
9 This implies that the h-index of the real-world instances is low and thus a promising
parameter. Unfortunately, 2-Club is W[1]-hard with respect to the h-index (Corollary 4.2.8).
123
4.7 Conclusion and Open Questions
We have settled the complexity status of 2-Club for most of the parameters
in the structural parameter space shown in Figure 4.2. On the theoretical
side, we extended existing fixed-parameter tractability results for the 2-Club
problem by providing polynomial kernels for the parameters cluster editing set
and feedback edge set. We further gave a direct algorithm for the parameter
treewidth and distance to cographs. Complementing these positive results, we
showed lower bounds on the kernel size for parameter vertex cover and on the
running time as well as on the kernel size for the dual parameter k 0 = n − `.
On the practical side, we provide the currently best implementation for
2-Club which, as demonstrated in the experiments, solves 2-Club in up to
five minutes even on large real-world graphs with more than 700,000 vertices.
Still, several open questions remain. First, there are obviously parameters
for which the parameterized complexity is still open. For example, is 2-Club
parameterized by distance to interval graphs or by distance to 2-club cluster
graphs in XP or even fixed-parameter tractable? In this context, also
parameter combinations could be of interest. Clearly a complete investigation
of the parameter space is infeasible. Hence, one should focus only on
practically relevant parameter combinations. One example could be the
following question that is left open by the hardness results for h-index and
degeneracy. Is 2-Club also W[1]-hard with respect to the parameter h-index
if the input graph has constant degeneracy? Second, it remains open whether
there is a polynomial kernel for the parameter distance to clique or to identify
further non-trivial structural parameters for which polynomial kernels exist.
Third, for many of the presented fixed-parameter tractability results it
would be interesting to either improve the running times or to obtain tight
lower bounds. For example, is it possible to solve 2-Club parameterized by
distance to clique in δ k · nO(1) time for some δ < 2? Similarly, is it possible
k
to solve 2-Club parameterized by vertex cover in 2o(2 ) · nO(1) time? An
answer to the latter question could be a first step towards improving the
(also doubly exponential) running time of the algorithms for the parameters
treewidth or distance to cographs. Concerning the parameter solution size `,
can the, so far impractical, running time or the size of the Turing kernel be
improved [Sch+12]? Finally, it would be interesting to see which results carry
over to 3-Club [Pas08, PB12] or to the related 2-Clique problem [BBT05].
4.8 Tables with Full Experimental Results
124
125
0
0.3
0.15
0.05
0.15
0.2
0
0.1
b
0.1
a
0
dens.
0.05
140
145
150
155
160
165
170
150
160
170
180
190
200
150
160
170
180
190
200
n
150
160
170
180
190
200
1459
1577
1672
1785
1920
2031
2155
1118
1261
1432
1618
1796
1989
1111
1279
1431
1606
1795
1981
m
556
632
723
801
896
989
39.5
41
42.2
43
44.8
46.1
47.4
27.2
28.6
29.9
32.1
33.3
35.3
30.1
32.2
34
35.4
37.4
39.5
∆G
17.7
18.4
19.5
20.8
21.6
22.2
20.8
21.8
22.3
23
24
24.6
25.4
14.9
15.8
16.9
18
18.9
19.9
14.8
16
16.8
17.8
18.9
19.8
∅G
7.41
7.9
8.51
8.89
9.43
9.89
27.4
28.6
29.3
30.4
31.6
32.3
33.3
19.8
20.9
22.2
23.4
24.5
25.8
21.2
22.6
23.9
25.2
26.7
28.1
h-index
12.5
13.1
13.9
14.6
15.5
16.2
71.5
77.4
80.3
84.2
91.5
94.4
99.5
P
28.2
29.6
30.9
33.1
34.3
36.3
P
31.2
33.5
35.4
37
39.1
41.8
P
2-club
18.7
19.4
20.5
21.8
22.6
23.2
P
4.8.1 Experimental Results for Random Instances
ST
0.08
0.1
0.12
0.14
0.16
0.2
0.80
0.82
1.7
2.99
6.45
13.3
25.6
50.9
0.7
1.18
2.54
4.87
8.8
15.2
33.3
0.68
0.65
0.86
1.09
0.95
1.68
1.77
7.7
ST-TK-DEG
0.57
0.74
0.92
1.17
1.41
1.71
6.52
0.88
1.79
3.13
6.78
13.8
26.2
52.6
0.73
1.23
2.61
5.02
9.03
15.5
34.1
0.69
0.65
0.86
1.09
0.95
1.69
1.79
7.7
ST-TK-N2
0.74
0.97
1.26
1.61
2.01
2.52
9.11
0.92
1.83
3.18
6.84
13.9
26.3
53.0
0.75
1.24
2.62
5.03
9.04
15.5
34.2
0.69
0.66
0.86
1.09
0.96
1.7
1.79
7.8
ST-TK-FES
0.84
1.11
1.43
1.82
2.29
2.85
10.34
0.97
1.88
3.23
6.89
13.9
26.4
53.3
0.77
1.27
2.65
5.05
9.07
15.6
34.4
0.71
0.67
0.88
1.11
0.98
1.72
1.82
7.9
CST
0.05
0.07
0.09
0.11
0.13
0.16
0.61
0.74
1.68
3.11
6.49
15.3
32.1
59.4
0.56
0.98
2.12
4.55
8.45
16.2
32.9
1.08
1.08
1.47
1.96
1.75
3.35
3.68
14.4
126
0.3
0.25
0.35
0.2
0.1
0.1
0.15
0.5
b
0.25
a
0.05
200
200
200
140
145
150
155
160
165
170
n
140
145
150
155
160
165
170
3976
3984
3970
1461
1563
1679
1787
1915
2032
2161
m
1462
1567
1666
1782
1913
2016
2151
61.3
57.9
65.8
34
34.8
35.8
37
38.2
39.8
40.1
∆G
36.1
37.6
38.7
40.2
41
41.9
43.6
39.8
39.8
39.7
20.9
21.6
22.4
23.1
23.9
24.6
25.4
∅G
20.9
21.6
22.2
23
23.9
24.4
25.3
45.8
44.6
47.7
25.1
25.8
26.8
27.5
28.6
29.3
30.3
h-index
26
27
27.7
28.8
29.8
30.5
31.5
191
195
185
P
50.7
53.6
57.7
59.9
65.5
68.5
74
P
2-club
60.4
64.5
67.5
72.2
78.9
81.4
87.4
P
ST
5.06
7.04
10.4
14.1
16.8
21.5
31
105.9
21.7
36.5
54.4
105
157
258
402
1035
0.02
0.02
0.02
0.1
ST-TK-DEG
5.1
7.1
10.5
14.2
17
21.7
31.3
106.9
22
37
55.1
106
158
260
405
1043
0.03
0.02
0.03
0.1
ST-TK-N2
5.11
7.12
10.5
14.2
17
21.7
31.3
106.9
22
37
55.1
105
158
260
405
1042
0.04
0.04
0.05
0.1
ST-TK-FES
5.13
7.13
10.5
14.2
17
21.7
31.3
107.0
22
37
55.1
105
158
260
405
1042
0.08
0.08
0.09
0.3
CST
7.15
10.7
17
25.2
30.8
41.1
64.2
196.2
30
54.1
85.5
171
272
500
834
1947
0.05
0.04
0.07
0.2
Table 4.4: Full list of experimental results on random instances with 0.05, 0.1, 0.15, and 0.2 density. For each combination
of a, b, and n we created 100 instances by the random graph generator proposed by Gendreau et al. [GSS93]. Correspondingly,
all other entries namely, m (# edges), ∆G (maximum degree), ∅G (avg. degree), h-index, 2-club (size of the largest 2-club),
and the time in seconds for the solvers ST, ST-TK-DEG, ST-TK-N2,ST-TK-FES, CST are the averages over all these 100 instances.
0.2
dens.
0.15
127
10th DIMACS
clustering
adjnoun
celegans_metabolic
celegansneural
dolphins
email
football
hep-th
jazz
karate
netscience
PGPgiantcompo
polblogs
polbooks
power
10th DIMACS
co-author citation
citationCiteseer
coAuthorsCiteseer
coAuthorsDBLP
graph_thres_01
graph_thres_02
graph_thres_03
graph_thres_04
graph_thres_05
2nd DIMACS clique
brock200_2
brock200_4
name
9876 114 98
13089 147 130
8
7
6
7
4
3
2
2
268495 1156647 1318
227320 814134 1372
299067 977676 336
715633 2511988 804
282831 640697 201
167006 293796 123
112949 168524 88
81519 107831 71
200
200
7
8
23
5
9
10
3
27
4
3
4
22
8
2
425
2025
3520
159
5451
613
15751
2742
78
2742
24316
16715
441
6594
50
238
285
13
72
16
51
103
18
35
206
352
28
20
0.01
0.10
0.03
0.00
42.8
0.07
0.94
0.05
0.00
0.03
1.14
9.78
0.01
0.25
99
128
200 0.04
200 0.03
209 1319 79.3
114 1373 49.4
132 337 85.0
208 805 294
96 202 62.4
62 124 25.7
46
89 14.2
38
72 8.93
13
23
34
9
33
12
27
41
6
19
52
87
15
12
82.5
49.8
91.6
284
60.6
25.6
14.3
8.91
0.01
0.10
0.03
0.00
110
0.38
0.93
0.05
0.00
0.03
1.15
44.9
0.03
0.25
0 0.10
0 0.15
0
0
0
0
0
0
0
0
0
0
0
0
171
5
0
0
0
0
0
4
3
0
78.2
49.5
85.2
275
59.5
25.5
14.3
8.97
0.00
0.11
0.03
0.00
83.2
0.46
0.94
0.07
0.00
0.03
1.15
21.0
0.07
0.25
0 0.20
0 0.26
0
0
0
0
0
0
0
0
0
0
0
0
293
66
0
0
0
0
0
42
10
0
78.7
48.8
86.7
292
61.3
25.0
13.9
8.74
0.06
0.24
0.00
0.11
0.03
0.01
70.5
2.73
0.93
0.23
0.00
0.03
1.14
0 0.92
0 0.86
0
0
0
0
0
0
0
0
0
0
0
0
234
68
0
0
0
0
0
12
17
0
0.01
0.10
0.03
0.01
140
0.77
0.93
0.24
0.00
0.03
1.16
78.5
48.7
85.0
287
59.9
25.1
13.9
8.70
0 0.99
0 0.97
0
0
0
0
0
0
0
0
5 0.08
0 0.24
0
0
0
0
162
4
0
0
0
0
0
0.00
0.11
0.03
0.01
112
0.87
0.95
0.26
0.00
0.03
1.15
82.0
49.6
85.3
277
59.2
25.2
13.9
8.77
0 1.08
0 1.09
0
0
0
0
0
0
0
0
11 0.14
0 0.25
0
0
0
0
293
66
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
18
0
0
0
0
0
234
68
0
0
0
0
0
m ∆G ∅G h-Index 2-club ST-TR-Deg ST-TR-N2 ST-TR-FES ILP-TR-Deg ILP-TR-N2 ILP-TR-FES
time #TK time #TK time #TK time #TK time #TK time #TK
49
237
284
12
71
12
50
100
17
34
205
351
25
19
112
453
297
62
1133
115
8361
198
34
1589
10680
1490
105
4941
n
4.8.2 Experimental Results for Real-World Instances
128
400
400
800
800
125
250
500
1000
500
200
200
400
400
400
256
171
776
378
1500
300
300
300
700
700
700
n
m ∆G ∅G h-Index 2-club ST-TR-Deg
time #TK
59786 328 298
294 400 0.26
0
59765 326 298
294 400 0.28
0
208166 566 520
516 800 2.17
0
207643 565 519
514 800 2.14
0
6963 119 111
107 125 0.01
0
27984 236 223
219 250 0.09
0
112332 468 449
441 500 0.63
0
249826 551 499
500 1000 2.24
0
62624 286 250
250 500 0.38
0
17910 190 179
173 200 0.04
0
17910 190 179
173 200 0.04
0
71820 375 359
347 400 0.31
0
71820 378 359
349 400 0.32
0
71820 380 359
349 400 0.31
0
20864 163 163
163 256 0.07
0
9435 124 110
106 171 0.02
0
225990 638 582
565 776 2.68
0
70551 374 373
364 378 0.30
0
284923 614 379
456 1500 26.8
0
10933 132 72
90 299 0.06
0
21928 229 146
148 300 0.09
0
33390 267 222
208 300 0.12
0
60999 286 174
207 700 0.74
0
121728 539 347
352 700 1.02
0
183010 627 522
486 700 1.70
0
ST-TR-N2 ST-TR-FES
time #TK time #TK
1.95
0 4.91
0
1.98
0 4.83
0
15.1
0 28.3
0
14.9
0 28.0
0
0.06
0 0.09
0
0.57
0 0.95
0
9.32
0 14.7
0
14.4
0 30.4
0
1.99
0 5.63
0
0.24
0 0.37
0
0.24
0 0.37
0
3.42
0 6.60
0
3.42
0 6.52
0
3.38
0 6.48
0
0.29
0 0.51
0
0.09
0 0.16
0
18.7
0 31.4
0
3.71
0 7.39
0
34.6
0 55.0
0
0.13
0 0.30
0
0.41
0 0.65
0
0.67
0 1.37
0
1.61
0 4.88
0
6.56
0 13.3
0
14.5
0 25.4
0
ILP-TR-Deg ILP-TR-N2
time #TK time #TK
5.90
0 7.64
0
4.78
0 6.62
0
96.8
0 110
0
83.1
0 96.0
0
0.08
0 0.13
0
0.81
0 1.24
0
5.55
0 13.8
0
181
0 194
0
15.7
0 17.5
0
0.39
0 0.60
0
0.40
0 0.59
0
2.70
0 5.92
0
2.68
0 5.85
0
2.68
0 5.92
0
2.25
0 2.30
0
0.68
0 0.74
0
74.7
0 91.3
0
0.48
0 3.58
0
288
0 296
0
1.50
0 1.47
0
2.76
0 3.01
0
1.88
0 2.47
0
21.7
0 22.7
0
49.2
0 55.4
0
33.7
0 46.7
0
ILP-TR-FES
time #TK
9.62
0
8.49
0
121
0
107
0
0.16
0
1.62
0
20.0
0
208
0
19.9
0
0.74
0
0.74
0
8.09
0
8.08
0
8.13
0
2.72
0
0.82
0
103
0
6.81
0
314
0
1.73
0
3.53
0
2.89
0
25.2
0
61.3
0
56.9
0
Table 4.5: Full list of experimental results on instances from the DIMACS implementation challenges [DIM12, DIM93]. The
first column denotes the name of the instances, n the number of vertices, m the number of edges, ∆G the maximum degree,
∅G the avg. degree, and 2-club denotes the size of the largest 2-club. The last twelve columns contain for each solver the
required time in seconds and “#TK” denotes the number of Turing kernels needed to solve.
brock400_2
brock400_4
brock800_2
brock800_4
C125.9
C250.9
C500.9
DSJC1000.5
DSJC500.5
gen200_p0.9_44
gen200_p0.9_55
gen400_p0.9_55
gen400_p0.9_65
gen400_p0.9_75
hamming8-4
keller4
keller5
MANN_a27
p_hat1500-1(1)
p_hat300-1
p_hat300-2
p_hat300-3
p_hat700-1
p_hat700-2
p_hat700-3
name
5 Parameterizing Local Search
for TSP by Neighborhood
Measures
We extend previous work [Mar08] on the parameterized complexity of local
search for the Traveling Salesman problem (TSP). So far, its parameterized complexity has been investigated with respect to the distance measures
(defining the local search area) Edge Exchange and Max-Shift. We perform
studies with respect to the distance measures Swap and r-Swap, Reversal
and r-Reversal, and Edit, achieving both fixed-parameter tractability and
W[1]-hardness results. In particular, from the parameterized reduction showing W[1]-hardness we infer running time lower bounds (based on the ETH,
Hypothesis 2.3.15) for all corresponding distance measures. Moreover, we
provide non-existence results for polynomial kernels and we show that some
in general W[1]-hard problems turn fixed-parameter tractable when restricted
to planar graphs.
5.1 Introduction
The NP-complete Traveling Salesman problem (TSP) [GJ79](ND22) is
probably the most studied combinatorial optimization problem. Almost all
algorithm design techniques have been applied to it or were even specifically developed for it [App+11, GP02, Law+85]. Famous results include the
Held/Karp-algorithm [HK62], the polynomial-time factor-1.5 approximation
algorithm for Metric TSP of Christofides [Chr76], and the polynomial-time
approximation scheme for Euclidean TSP [Aro98]. Many heuristic algorithms for TSP have been developed and evaluated [JM04, JM97]. Most of
them follow the paradigm of local search: Incrementally try to improve a solution by searching within its local neighborhood defined by a distance measure.
Perhaps the most prominent and best examined distance measure for TSP is
the k-Edge Exchange neighborhood (also called k-Opt neighborhood in some
literature), where one is allowed to exchange at most k edges of the Hamiltonian cycle forming the tour. Implementations of this strategy for k = 2, 3
and generalizations such as the Lin-Kernighan-heuristic [LK73] belong to the
129
best performing heuristic algorithms for real-world instances [JM04] both
in terms of quality and in terms of running time. However, for larger k,
for which one would expect a strong increase of quality, the running time
becomes infeasible since until now no algorithm is known which significantly
beats the trivial O(nk ) running time needed for a brute-force exploration
of the local distance-k neighborhood on n-vertex graphs. In an important
step forward, considering the problem within the framework of parameterized complexity, Marx [Mar08] has shown by proving W[1]-hardness that
there is no hope for fixed-parameter tractability with respect to k in case of
k-Edge Exchange neighborhood. Moreover, assuming that the ETH (Hypothesis 2.3.15) does not
fail, Marx [Mar08] has shown that there is no algorithm
√
3
running in O(no( k) ) time.
In this work, besides the k-Edge Exchange neighborhood (briefly, Edge
distance measure), we consider various other distance measures such as the
Reversal distance where the order of some consecutive vertices is reversed,1
the Swap distance where one is allowed to exchange two vertices, and the
Edit distance where one can move a vertex to an arbitrary new position.
We describe a round-tour in a graph via a permutation of its vertices and
each of these permutations naturally corresponds to a Hamiltonian cycle,
that is, a cycle in a graph containing all vertices. A distance measure maps
any two Hamiltonian cycle to a natural number describing how “close” they
are. For λ being any of the distance measures mentioned above (e. g. Swap
distance), we study the following problem.
LocalTSP(λ)
Input: An undirected graph G = (V, E) with vertices labeled v1 , . . . , vn
such that the identical permutation (id) v1 , v2 , . . . , vn , v1 is a
Hamiltonian cycle in G, an edge weight function ω : E → R+
0,
and a positive integer k.
Question: Is there a permutation π with λ(π, id) ≤ k that yields a Hamiltonian cycle with ω(π) < ω(id)?
In the definition above the weight ω(π) of any Hamiltonian cycle π is
Pn−1
defined to be ω(π) = i=1 ω({vπ(i) , vπ(i+1) }) + ω({vπ(n) , vπ(1) }) (the total
weight sum of all edges on the Hamiltonian cycle). Reflecting different distance measures λ, we speak about LocalTSP(Edge), LocalTSP(Reversal),
LocalTSP(Swap), LocalTSP(Edit), etc. We use LocalTSP if the measure in use is not important. Notably, all problems have the same set
of instances.
1 The
reversal distance is also widely studied in bioinformatics in the context of genome
rearrangements [Cap97, Fer+09].
130
Table 5.1: Overview of our results using k as the parameter and assuming r to be
a constant. The two results written in italics are a direct consequence of a more
general result. Furthermore, unless NP ⊆ coNP/poly, we show that for all distance
measures in the table there cannot be a polynomial kernel even on planar graphs
(Theorem 5.4.4). Results marked by ∗ indicate that the corresponding algorithm is
only a permissive parameterized algorithm (see Section 5.2).
r-Swap
Swap
Edit
r-Reversal Reversal Edge
general
FPT
W[1]-h
W[1]-h
FPT
W[1]-h W[1]-h
graphs (Thm. 5.3.6) (Thm. 5.3.1) (Thm. 5.3.1) (Thm. 5.3.7) (Thm. 5.3.1) [Mar08]
planar
FPT
FPT∗
FPT∗
FPT
?
?
graphs
(Thm. 5.4.8) (Thm. 5.4.8)
5.1.1 Related Work
The most important reference point to our work are Marx’s [Mar08] results on LocalTSP(Edge) (using different notation). Long before Marx,
Balas [Bal99] studied LocalTSP(Max-Shift), where Max-Shift distance k
means that in order to obtain an improved Hamiltonian cycle the maximum number of positions that a vertex is allowed to shift is k. Contrasting
the parameterized hardness result of Marx [Mar08], Balas showed that LocalTSP(Max-Shift) is fixed-parameter tractable by providing an algorithm
running in O(4k−1 k 1.5 n) time.
5.1.2 Our Contribution
Table 5.1 summarizes our results. Following the third approach to identify
tractable cases of NP-hard problems (Chapter 1), we performed a systematic
parameterized complexity study of different neighborhood measures. Most
importantly for this is a “hierarchical structure” of the distance measures
which similar to the relation of being a stronger parameterization (Definition 2.3.14). We describe the structure of the corresponding parameter space
in detail in Section 5.2.
We show that the W[1]-hardness result due to Marx [Mar08] for LocalTSP(Edge) can be extended, that is, we show that LocalTSP(λ) for
λ ∈ {Swap, Edit, Reversal} is also W[1]-hard, implying that it is probably
not fixed-parameter tractable for the “locality parameter” k. Furthermore, we
strengthen Marx’s running time lower bound based on the ETH by showing
that LocalTSP(λ) for λ ∈ {Swap, Edit, Reversal, Edge} does not admit
an algorithm with running time O(no(k/ log k) ).
131
Continuing to chart the border of tractability in the parameter space of
distance measures, for the Swap distance we show that, restricting by a
parameter r the distance of two vertices that are allowed to swap, makes
LocalTSP(r-Swap) fixed-parameter tractable with respect to the combined
parameter (k, r). Specifically, we outline an algorithm running in O(r2k (r −
1)2k · 4k · (k 2 + n) · n) time. Furthermore, we show that an analogously
restricted Reversal distance, called r-Reversal, admits an algorithm running
in O(2rk · r2k−1 · (r − 1)k · (k 2 + rk + n) · n) time and thus again leads to
fixed-parameter tractability.
Having shown that, except r-Swap/Reversal, none of the distance measures
under consideration admit a parameterized algorithm on general graphs we
combine the approach to examine different local search distance measures
with the first approach to identify tractable cases of NP-hard problems (consider special graph classes). More specifically, motivated by the question of
Fellows et al. [Fel+12] on the parameterized complexity of LocalTSP(Edge)
on planar graphs we study LocalTSP on planar graphs. We show that
LocalTSP(λ) for λ ∈ {Swap, Edit} is fixed-parameter tractable on planar
graphs. In addition, exploring the limitations of kernelization, we indicate
that, unless NP ⊆ coNP/poly, even on planar graphs there is no polynomial
kernel for LocalTSP(λ) for any of the considered distance measures λ.
5.2 Preliminaries: Parameterized Local Search
and Distance Measures
Local Search in Parameterized Algorithmics. It is very natural to use
parameterized algorithmics to study the computational complexity of local
search measured in the size of the local neighborhood where one tries to find
an improved solution. In fact, when k measures the “diameter” of the local
neighborhood, it is often not hard to come up with an algorithm running
in nO(k) time, but since such an algorithm usually becomes intractable
already for very small k, the question whether there is an algorithm with
running time f (k)·nO(1) for a moderately growing function f naturally arises.
Parameterized algorithmics provides a framework to prove the existence of
such algorithms or to deliver some evidence that it cannot exist.
We will briefly summarize the state of the art on parameterized results for local search. Fellows et al. [Fel+12] showed that searching the
k-exchange neighborhood for problems such as r-Center, Vertex Cover,
Odd Cycle Transversal, Max-Cut, and Min-Bisection can be done
on planar graphs in 2O(k) · n2 time, and is W[1]-hard on general graphs.
Fomin et al. [Fom+10] outlined a color-coding based algorithm for Weighted
132
Feedback Arc Set in Tournaments that decides in O(2o(k) · n log n)
time whether there is an improved solution in the k-exchange neighborhood
(symmetric difference of the corresponding arc sets). Marx and Schlotter [MS11] studied a variant of the Stable Marriage problem with respect to local search in the framework of parameterized algorithmics. To
analyze local search in the framework of parameterized algorithmics is
relatively new; further applications include Boolean Constraint Satisfaction [KM12], Incremental Coloring [HN13b], Sat [Sze11], and
Vertex Cover [Gas+12].
Permissive Algorithms. Marx and Schlotter [MS11] proposed to distinguish
between strict and permissive local search algorithms. Strict local search
algorithms find an improved solution (or prove that it does not exist) within
some limited distance from the given solution. Permissive local search
algorithms are allowed to find any improved solution (potentially, with
unbounded distance to the given solution). (Clearly, when no improved
solution within the limited distance exists a permissive algorithm is allowed
to answer no.) The motivation for this distinction is that finding an improved
solution within a bounded distance of a given solution may be hard even
for problems where an optimal solution can easily be found, e. g., Minimum
Vertex Cover on bipartite graphs [KM12]. Gaspers et al. [Gas+12] recently
showed for Vertex Cover, which is NP-hard on 2-subdivided graphs, that
on 2-subdivided graphs strict local search is W[1]-hard while permissive local
search is fixed-parameter tractable.
Notation. Let Sn denote the set of all permutations on {1, . . . , n} and
let id ∈ Sn be the identity. If not otherwise stated, we consider undirected
simple graphs G = (V, E) with vertices labeled v1 , v2 , . . . , vn (see Section 2.1
for our graph-theoretic notation). A Hamiltonian cycle through such a
graph is expressed by a permutation π ∈ Sn such that the edge set E(π)
of π, defined as E(π) = {{vπ(i) , vπ(i+1) } | 1 ≤ i < n} ∪ {{vπ(n) , vπ(1) }}, is
a subset of E. In case of a directed graph G = (V, A) we require that A
contains (vπ(i) , vπ(i+1) ) for all i < n and (vπ(n) , vπ(1) ). For a weight function
P
ω : E → R+
0 we define the weight of π by ω(π) =
e∈E(π) ω(e). The
Hamiltonian cycle π is called improved compared to id when ω(π) < ω(id).
In this sense, LocalTSP(λ) is the question whether there is an improved
Hamiltonian cycle π with λ(π, id) ≤ k.
Distance Measures. So far, the distance between Hamiltonian cycles was
usually measured in terms of Edge distance, counting the number of edges
used by one cycle but not used by the other. Another measure considered is
133
the Max-Shift distance, which equals the maximum shift of the position of a
vertex between the two permutations [Bal99]. We consider several further
measures based on the following operations on permutations.
Definition 5.2.1 (Distance Measures). For a permutation 1, 2, . . . , n, we
define the following operations:
reversal ρ(i, j)
results in
1, . . . , i − 1, j, j − 1, . . . , i + 1, i, j + 1, . . . , n;
swap σ(i, j)
results in
1, . . . , i − 1, j, i + 1, . . . , j − 1, i, j + 1, . . . , n;
edit (i, j)
results in
1, . . . , i − 1, i + 1, . . . , j − 1, j, i, j + 1, . . . , n.
For a constant r a swap σ(i, j) (or a reversal ρ(i, j)) is called an r-swap
(r-reversal, resp.) if 0 < j − i ≤ r − 1 or n + j − i ≤ r − 1. The distance
measures Swap, r-Swap, Edit, Reversal, and r-Reversal count the minimum
number of the appropriate operations to apply to one permutation in order
to obtain the other.
We do not consider the elements 1 and n to be anyhow special and,
therefore, the operations above can also be applied “over them”, e. g. σ(n−1, 2)
is a 4-swap.
We next show how the relation between the distance measures from
Definition 5.2.2 can be used to easily transfer results shown for one distance
measure to other ones.
Definition 5.2.2 (Boundedness of Distance Measures). A distance measure λ is called bounded by a distance measure τ (or τ -bounded) if there is a
function f : N → N such that for any two permutations π, π 0 ∈ Sn it holds
that λ(π, π 0 ) ≤ f (τ (π, π 0 )).
Similar to the notation of stronger parameterizations (Definition 2.3.14),
the relation of boundedness is reflexive and transitive and, therefore, forms
a quasi-order on the distance measures. Thus, the relation of boundedness
defines our parameter space on distance measures. Figure 5.1 depicts all
relations between the measures, omitting relations that can be deduced from
the transitivity, in this sense showing a Hasse diagram of this quasi-order.
Before arguing about correctness of the relations depicted in Figure 5.1, we
show a tight relationship between our notion of bounded distance measures
and the existence of permissive parameterized algorithms (a parameterized
algorithm for the permissive local search variant).
Lemma 5.2.3. If a distance measure λ is τ -bounded, then a (permissive)
parameterized algorithm for LocalTSP(λ) is a permissive parameterized
algorithm for LocalTSP(τ ).
134
2k
Edge
Reversal
2k
3k
2k
Edit
2k
Swap
Max-Shift
rk
k
rk
2
r-Swap
rk
2
rk
r-Reversal
2k
Figure 5.1: Hasse diagram of the relations between the distance measures. Let
f : N → N. An arrow labeled “f (k)” from a distance measure τ to a measure λ
means that λ is τ -bounded with function f (k), implying that two Hamiltonian
cycles of distance k with respect to τ have distance at most f (k) with respect to λ.
Proof. Consider an instance of LocalTSP(τ ) with an improved Hamiltonian
cycle in τ -distance at most k from the given Hamiltonian cycle. Then this
improved Hamiltonian cycle is in λ-distance at most f (k) from the given
Hamiltonian cycle. Thus, running the (permissive) parameterized algorithm
for LocalTSP(λ) with parameter f (k) returns an improved Hamiltonian
cycle and thus is a permissive algorithm for LocalTSP(τ ).
We next argue about the correctness of the relations depicted in Figure 5.1
(we consider r to be a constant in these comparisons): Obviously, an r-swap is
a special case of a swap and, therefore, Swap distance is bounded by r-Swap
distance. Next, an r-swap can be simulated by at most two r-reversals and a
swap can be simulated by two edits. Thus, the r-Reversal distance is bounded
by the r-Swap distance and the Edit distance is Swap-bounded. Further,
one r-swap or r-reversal shifts a position of any vertex in the Hamiltonian
cycle by at most r, and, therefore, k of them shift no vertex by more than
rk, which implies that Max-Shift distance is both r-Swap-bounded and
r-Reversal-bounded. Similarly an r-reversal can be simulated by at most r/2
r-swaps and, hence, r-Swap distance is r-Reversal-bounded. Since one edit
can be simulated by at most two reversals and a reversal breaks at most two
edges, it follows that Reversal distance is Edit-bounded and Edge distance
135
Figure 5.2: A planar graph with two different Hamiltonian cycles (marked by bold
lines). The cycles are only four edge modifications and four reversals from each
other, while they can be made arbitrarily far apart for any other of the measures by
extending the horizontal lines. Furthermore, the vertices affected by the changes
are also arbitrarily large apart from each other in the underlying graph and, as
the graph has no other Hamiltonian cycles, there is no other solution with changes
concentrated in a constant distance to one particular vertex.
is Reversal-bounded. Additionally, an edit breaks at most three edges and
thus the Edge distance is Edit-bounded. It remains to show that Reversal
distance is Edge-bounded.
Lemma 5.2.4. The Reversal distance is bounded by the Edge distance.
Proof. Assume that from a given Hamiltonian cycle one can obtain an improved one by first deleting k edges and adding another k edges. Consider the
paths of the given Hamiltonian cycle after we remove k edges. Now we build
the improved Hamiltonian cycle by gradually connecting the appropriate
paths. We start from any path and consider the path that should come
next. By two reversals we can achieve that the paths follow each other in
the correct order (the first reversal moves the path next to the previous one
and the second rotates it into the right direction). By this we introduce at
least one edge of the new Hamiltonian cycle. As we never break any edge
not deleted from the Hamiltonian cycle and the reversals can be taken to
operate outside the already built part of the Hamiltonian cycle, we build the
whole improved Hamiltonian cycle by at most 2k reversals. This shows that
Reversal distance is bounded by Edge distance.
It is also not hard to come up with examples showing that no further
boundedness relations hold between the distance measures. See Figure 5.2
for an interesting case of two Hamiltonian cycles which are close for Reversal
and Edge distances, but far apart for all the other distances considered.
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5.3 Parameterized Complexity of LocalTSP on
General Graphs
In this section, we provide parameterized hardness as well as fixed-parameter
tractability results for LocalTSP using various distance measures.
5.3.1 W[1]-Hardness and an O(no(k/ log k) ) Lower Bound
We show that LocalTSP(λ) is W[1]-hard for λ ∈ {Swap, Edit, Reversal}.
Furthermore, for all these distance measures plus the Edge distance we provide a computational lower bound of O(no(k/ log k) ). To this end, we build on
the W[1]-hardness proof for LocalTSP(Edge) by Marx [Mar08]. In contrast
to Marx, who gave a parameterized reduction from the k-Clique problem,
we reduce from the Partitioned Subgraph Isomorphism problem. This
makes the construction more structured and more powerful.
Partitioned Subgraph Isomorphism (PSI)
Input: Two undirected graphs H and G with |V (H)| ≤ |V (G)|, and a
(not necessarily proper) coloring f : V (G) → V (H) of vertices
of G with vertices of H.
Question: Is there a mapping h : V (H) → V (G) such that ∀v ∈ V (H) :
f (h(v)) = v and h is a homomorphism, that is, ∀{u, v} ∈
E(H) : {h(u), h(v)} ∈ E(G)?
If such a homomorphism h exists, then we say that there is a colored
H-subgraph in G. PSI is W[1]-hard for the parameter k = |E(H)| as it is a
generalization of the W[1]-hard k-Multicolored Clique problem [Fel+09].
Our main result in this section is the following.
Theorem 5.3.1. LocalTSP(λ) is W[1]-hard with respect to k for λ ∈
{Swap, Edit, Reversal, Edge}.
Theorem 5.3.1 follows from the following lemma, the fact that PSI is
W[1]-hard and from the fact that the Edit, Reversal, and Edge distances are
all Swap-bounded.
Lemma 5.3.2. There is a parameterized reduction from Partitioned Subgraph Isomorphism parameterized by k = |E(H)| to LocalTSP(Swap)
such that any improved Hamiltonian cycle can be obtained by performing at
most O(k) swaps.
Proof. We provide a parameterized reduction from PSI parameterized by
k = |E(H)| to DirectedLocalTSP(Swap), this is, the variant of LocalTSP(Swap) where the input graph is directed. We show that in the
137
α
β
α
β
α
β
γ
δ
γ
δ
γ
δ
(a)
(b)
(c)
Figure 5.3: Switch gadget (a) and the two possible ways to traverse it (b) and (c).
constructed graph any improved Hamiltonian cycle can be obtained by performing at most 24k swaps to the given cycle. The claim is then obtained
from the parameterized reduction from DirectedLocalTSP(Swap) to
LocalTSP(Swap) given by Marx [Mar08, Lemma 3.1]. Since the reduction is also polynomial-time computable it is indeed a linear-parameter
transformation.
Construction: Assume that the graph G = (V, E) with V = {v1 , . . . , vn },
the graph H with k = |E(H)| and V (H) = {1, . . . , l}, and a coloring f constitute an instance of PSI. We assume without loss of generality that H is connected. We construct an equivalent instance of DirectedLocalTSP(Swap)
on the directed graph D by multiple copies of the so-called switch gadget
(see Figure 5.3(a)). There are only two possibilities to traverse a switch on a
Hamiltonian cycle, by using either the upper path α → β (Figure 5.3(b)) or
the lower path γ → δ (Figure 5.3(c)). Next, since the constructed graph D
will contain only one non-zero-weight arc, all arcs have weight zero if not
explicitly stated otherwise.
Each vertex vi ∈ V is represented in D by its segment Vi , which is formed
by degH (f (vi )) many switch gadgets Vi,j , where j ∈ {1, . . . , degH (f (vi ))}.
To form the segment, we sequentially connect the switches by connecting
each β-vertex to the α-vertex of the subsequent switch. Furthermore, we
add a start vertex vis which has an outgoing arc to the α-vertex of the first
switch and an end vertex vie which has an incoming arc from the β-vertex of
the last switch.
We connect all segments V1 , V2 , . . . , Vn by adding an arc from the end
s
vertex vie of segment Vi to the start vertex vi+1
of segment Vi+1 for all
s
1 ≤ i < n. In addition, we add a start vertex v which has an outgoing arc
to the start vertex v1s of the first segment V1 and an end vertex v e which has
an incoming arc from the end vertex vne of the last segment Vn .
For each color j ∈ V (H) we add a so-called template tj , which is a directed
138
simple path consisting of 6 degH (j) vertices. Furthermore, for all segments Vi
where f (vi ) = j there is an arc from vis to the first template vertex tsj and
an arc from the last template vertex tej to vie . Finally, there are arcs from v e
to ts1 , from tej to tsj+1 for all j ∈ V (H), and an arc of weight one from the
last vertex in the last template tel to v s . This allows to traverse all templates
starting from the end vertex v e , and after reaching tel we can go back to the
start vertex v s .
The given Hamiltonian cycle C in our DirectedLocalTSP(Swap) instance on D is as follows. It starts in v s , traverses each segment by using the
upper path for each switch, follows the zero-weight arcs between subsequent
segments, ends up in v e , afterwards traverses all templates and finally uses
the weight-one arc from tel to get back to the start vertex v s . Hence, the
weight of cycle C is one.
Observe that, once a cycle enters a segment Vi via vie and begins to traverse
the first switch by using the upper path, it has to traverse all switches in this
segment by the upper path. We call such a segment passive. Symmetrically,
if one switch is traversed by the lower path, all switches in the same segment
have to be traversed by the lower path, and we call such a segment active.
All segments on the cycle C are passive and, so far, it is the only possible
Hamiltonian cycle through D.
Now, by adding some further arcs to D we will “encode” the structure
of G to ensure the existence of an alternative cycle C 0 with weight strictly
less than w(C) = 1 if and only if there is a colored H-subgraph in G. The
only possibility to get a cycle with weight less than w(C) = 1 is to skip the
weight-one arc (tel , v s ). The idea is that the cycle C 0 starts in v s , follows the
order of the segments but (in distinction to C) “decides” for each segment Vi
whether it traverses the segment through the switches or it “skips” the
segment by using the template tf (vi ) . After the cycle reaches v e , the new
arcs allow C 0 to traverse all skipped segments by using the lower path for
each switch, and in this way all these switches become active. The vertices
in G which correspond to the active segments in D on C 0 are intended to
form a colored H-subgraph.
For the purpose of a formal description of the new arcs in D which
are necessary to traverse the active segments starting at v e , consider the
directed graph H 0 obtained from H by replacing each edge by two arcs
with opposite directions. Further, consider a closed Eulerian cycle p0 =
p0 , p1 , p2 . . . , p2k−1 , p2k in H 0 , where pi ∈ V (H 0 ) = V (H), p0 = p2k , and
(pi−1 , pi ) ∈ A(H 0 ) for every i ∈ {1, . . . , 2k}. We remove p0 from p0 to obtain
p = p1 , . . . , p2k . In the trail p, we define b(pi ) = |{i0 | i0 ≤ i and pi0 = pi }|
to count how many times vertex pi already appeared on p when we reach pi
(we have b(pi ) ∈ {1, . . . , degH (pi )}).
We want our new cycle C 0 to follow the Eulerian trail p, using the b(pi )-th
139
switch of the segment of some vertex of color pi in the i-th place. To allow
this, we add arcs between the segments as follows. If there is an edge
{vi , vj } ∈ E(G) and the arc (f (vi ), f (vj )) appears in p as (pφ , pφ+1 ), then
add into D the arc from the δ-vertex of the switch Vi,b(pφ ) to the γ-vertex of
switch Vj,b(pφ+1 ) . To complete the construction of D, connect this structure
to the rest of the graph by adding for each vertex vi with f (vi ) = p1 an arc
from ve to the γ-vertex of switch Vi,1 and by adding for each vertex vj with
f (vj ) = p2k an arc from the δ-vertex of the switch Vj,b(p2k ) to the vertex v s .
Correctness: We show that there is a Hamiltonian cycle C 0 of weight
zero within Swap-distance at most 24k of C if and only if there is a colored
H-subgraph in G. Furthermore, we show that any improved Hamiltonian
cycle in D can be obtained by performing at most 24k swaps.
“⇐”: Assume that the vertices vi1 , vi2 , . . . , vil ∈ V (G) form a colored
H-subgraph in G such that f (vij ) = j for all j ∈ V (H). Then there is a
cycle C 0 that, starting in v s , uses the template tj instead of segment Vij ,
continues after reaching v e to the γ-vertex of switch Vi1 ,1 , then mimics the
trail p and finishes by using the arc from the δ-vertex of switch Vil ,b(p2k ) to
vertex v s . Note that each switch is traversed exactly once. Moreover, for
each j ∈ V (H) there is exactly on active segment Vi corresponding to the
vertex vi ∈ V (G) with f (vi ) = j.
It remains to count the number of swaps that have to be performed to C
to get C 0 . To this end, observe that the template of color j as well as the
switches of a segment (without start and end vertex) of a vertex of this
color consists of exactly 6 degH (j) vertices each. Since activating a segment
is done by swapping the vertices in the corresponding switches with the
vertices in a template, we only need
P 6 degH (j) swaps to activate a segment of
color j. In total we need exactly j∈V (H) 6 degH (j) = 12k swaps to activate
all segments. Finally, we need at most 12k swaps to “sort” the vertices in
the active segments such that the order of the switches corresponds to the
trail p. We can conclude that the Hamiltonian cycle C 0 can be obtained
from C by performing at most 24k swaps.
“⇒”: For the reverse direction, assume that in D there is a cycle C 0 of
weight zero. We prove that C 0 is within Swap-distance 24k from C and that
there is a colored H-subgraph in G. As cycle C 0 has to be different from C,
at least one segment Vu has to be active in it. Call a color j active if there
is a vertex vi ∈ V with f (vi ) = j such that the segment Vi is active and
passive otherwise.
We first show that in C 0 all colors are active. Towards a contradiction
assume there is a passive color x. Then there must be two colors x0 and y 0 such
that x0 is passive, y 0 is active, and {x0 , y 0 } ∈ E(H), since there is at least one
active color and H is connected. The trail p uses (y 0 , x0 ) or (x0 , y 0 ). Consider
the first case, that is, y 0 = pi and x0 = pi+1 for some i ∈ {1, . . . , 2k − 1}.
140
Also suppose that Vu0 is an active segment with f (vu0 ) = y 0 . Due to our
construction, the only way how the cycle C 0 can leave the switch Vu0 ,b(pi ) is
that there is an active segment corresponding to a vertex of color x0 , which is
a contradiction. As the second case also leads to a contradiction in a similar
way, we get that every color is active.
On the other hand, there is at most one active segment of each color, as the
only way to make a segment active is to replace it by the template. Hence,
for each color j there is exactly one vertex vij such that the segment Vij is
active. Since, in order to traverse these active segments, the Hamiltonian
cycle C 0 has to follow the Eulerian trail p, this enforces that the vertices vij
and vij0 are adjacent in G whenever {j, j 0 } ∈ E(H) and thus vij ’s form a
colored H-subgraph in G.
Moreover, since the improved Hamiltonian cycle C 0 contains exactly one
active segment for each vertex in V (H), it follows that C 0 is within Swapdistance 24k of C: As already argued above, replacing |V (H)| many segments
in the cycle C by the corresponding templates and then sorting the switches
within these active segments according to the Eulerian trail p is possible by
at most 24k swaps.
We next show the running time lower bound that can be derived from
Lemma 5.3.2. To this end, the following theorem of Marx [Mar10], proving
a lower bound for PSI, is extremely useful.
Theorem 5.3.3 ([Mar10, Corollary 6.3]). Unless the ETH fails, Partitioned Subgraph Isomorphism cannot be solved in f (H) · no(k/ log k) time,
where f is an arbitrary function and k = |E(H)|.
Lemma 5.3.2 together with Theorem 5.3.3 implies the following√corollary.
3
For the case of Edge distance, it improves the lower bound O(no( k) ) given
by Marx [Mar08, Corollary 3.5].
Corollary 5.3.4. Unless the ETH fails, LocalTSP(λ) does not admit a
(permissive) algorithm with running time O(no(k/ log k) ) for λ ∈ {Swap, Edit,
Reversal, Edge}.
5.3.2 Tractability
In Section 5.3.1 we have shown that on general graphs LocalTSP(λ) for
λ ∈ {Swap, Edit, Reversal, Edge} is W[1]-hard. In this section we show
that LocalTSP becomes fixed-parameter tractable when using the more
restrictive distance measures r-Swap and r-Reversal instead of Swap and
Reversal, respectively. Actually, we prove a stronger result, that is, fixedparameter tractability with respect to the combined parameter (k, r). The
141
corresponding algorithms are based on the bounded search tree technique,
and they are mainly based on the observation that the solution can be
assumed to be given by a sequence of r-swaps (or r-reversals, resp.) that are
somehow related.
We first describe the algorithm for LocalTSP(r-Swap). To this end, we
need the following definition. Let S be a sequence of swaps. We define an
undirected auxiliary swap graph GS as follows. There is a vertex for each
swap in the sequence S, and two swaps σ(i, j) and σ(t, l) are adjacent if
either t or l is contained in {i − 1, i, i + 1, j − 1, j, j + 1}. Furthermore, if a
swap σ(i, j) is applied, then we call the positions i and j and the vertices at
these positions affected.
Lemma 5.3.5. If a LocalTSP(λ) instance for λ ∈ {r-Swap, Swap} admits
an improved Hamiltonian cycle, it also admits an improved Hamiltonian
cycle which can be obtained by swaps (or r-swaps) such that their swap graph
is connected.
Proof. Suppose that we are given a sequence S of swaps whose application
to the Hamiltonian cycle id ∈ Sn creates an improved Hamiltonian cycle π ∈
Sn . Towards a contradiction, assume that C1 , . . . , Cp with p ≥ 2 are the
connected components of the corresponding swap graph GS . For any of these
components C, we denote by π C ∈ Sn the permutation that results from
applying the swaps in C to id preserving their order relative to S.
We shall show that the sets E(π C1 ) 4 E(id), . . . , E(π Cp ) 4 E(id) form
a partition of the set E(π) 4 E(id) (4 denotes the symmetric difference).
Having proved this, the rest of the argumentation is as follows. Since ω(π) <
ω(id) or equivalently ω(E(π) \ E(id)) < ω(E(id) \ E(π)), it follows that there
is at least one component C of GS with ω(E(π C )\E(id)) < ω(E(id)\E(π C )).
This implies that ω(π C ) < ω(id) and thus applying only swaps contained
in C also results in an improved Hamiltonian cycle π C .
It remains to prove that E(π C1 ) 4 E(id), . . . , E(π Cp ) 4 E(id) is a partition
of E(π) 4 E(id). First, for all 1 ≤ i < j ≤ p, by definition of the swap
graph it follows that the positions, and thus also the vertices, affected
by Ci are disjoint from the positions and vertices that are affected by Cj .
Formally, (E(π Ci ) 4 E(id)) ∩ (E(π Cj ) 4 E(id)) = ∅. For any component C,
we next argue that E(π C ) 4 E(id) ⊆ E(π) 4 E(id). Clearly, for an edge
e = {i, j} ∈ E(π C ) 4 E(id), either vertex i or j has to be affected by at least
one swap in C. Then, no swap in S \ C affects any of i and j, because such a
swap would be adjacent to at least one swap in C. Hence, e ∈ E(π) 4 E(id).
Finally, consider an edge e = {i, j} ∈ E(π)4E(id). By the same argument
as above, all swaps that affect any of i and j belong to the same component
of GS . Thus, since either vertex i or j is affected by a swap, it follows that
there is a component C of GS such that e ∈ E(π C ) 4 E(id).
142
The connectedness of the swap graph which is provided by Lemma 5.3.5
forms the basis of our parameterized algorithm for LocalTSP(r-Swap).
Theorem 5.3.6. LocalTSP(r-Swap) is fixed-parameter tractable with respect to the combined parameter (k, r). It is solvable in O(r2k (r − 1)2k · 4k ·
(k 2 + n) · n) time.
Proof. Let (G, ω, k) be an instance of LocalTSP(r-Swap). Furthermore,
let S be a sequence of at most k r-swaps such that applying S to id results in
an improved Hamiltonian cycle π. By Lemma 5.3.5 we can assume that GS is
connected. The algorithm consists of two parts. First, the algorithm guesses
the positions of all swaps in S and, second, it finds their correct order.
To describe the first part, for convenience, we assume for all swaps σ(i, j)
that j ∈ {i + 1, i + 2, . . . , i + r − 1}. Furthermore, we define an ordering
relation ≤ on swaps with σ(i, j) ≤ σ(t, p) if and only if i < t or i = t ∧ j ≤ p.
Let σ1 , σ2 , . . . , σs with s ≤ k be the swaps of S sorted with respect to ≤ in
ascending order.
In the first part of the algorithm, by branching into all possibilities for the
positions of the swaps, the algorithm guesses all swaps in the order given
above: At the beginning, the algorithm branches into all possibilities to
find the position i1 for σ1 (i1 , j1 ) and then into the r − 1 possibilities to find
the position j1 . Now, suppose we have already found the swap σt (it , jt ).
We next describe how to find the swap σt+1 (it+1 , jt+1 ). By the ordering
we know that i1 ≤ . . . ≤ it ≤ it+1 and, since all swaps are r-swaps, for all
1 ≤ p ≤ t with jp > it it holds that jp − it ≤ r − 1. From this and since
GS is connected (Lemma 5.3.5), it follows that it+1 − it ≤ r. Thus, we can
find the position of it+1 by branching into r + 1 possibilities. Afterwards,
by branching into r − 1 possibilities we find the position jt+1 . Overall, the
positions of σt+1 can be guessed by branching into at most r2 possibilities,
and there are at most r2k−1 · n possible positions of the swaps in total.
In the second part, the algorithm guesses the order of the r-swaps. Clearly,
the trivial way to do that is by trying all permutations of the swaps, resulting
in a total running time of O(r2k−1 k!·n). This already shows that the problem
is fixed-parameter tractable for (k, r). We next describe how this can be
accelerated in case that 4r2 < k. To this end, let σ (1) , σ (2) , . . . , σ (s) be all
swaps in S in the order of their application resulting in π. Clearly, if there are
two subsequent swaps σ (t) (i, j) and σ (t+1) (i0 , j 0 ) such that {i, j}∩{i0 , j 0 } = ∅,
then reversing their order in the application of the swaps also results in π.
More generally, instead of finding a total order of the swaps, it is sufficient
to find a partial order of the swaps that defines the order for any pair of
swaps σ(i, j) and σ(t, p) where |{i, j} ∩ {t, p}| = 1. Clearly, we do not have
to define the order of two swaps which are of the same emphtype, that is,
where {i, j} = {t, p}. Thus, for a position i, consider all swaps which affect
143
position i. Since all these swaps are r-swaps, there can be at most 2r − 2
different types that affect position i. Hence, if there are ki swaps that affect
position i, then there are at most (2r − 2)ki different permutations of these
swaps. Combining the number of possibilities of all affected positions, since
each swap affects exactly two positions, it follows that there are at most
(2r − 2)2k permutations of all swaps yielding different Hamiltonian cycles.
Once the partial orders at all relevant positions are determined, we check
whether this can be obtained by some total order of the swaps, and find this
order in O(k 2 ) time, by representing the partial orders by arcs in a directed
graph on the set of swaps and finding a topological order for this graph.
Then we apply the swaps in this order in O(k) time and check whether we
obtain an improved Hamiltonian cycle in linear time. Together with the first
part, the whole algorithm runs in O(r2k (r − 1)2k · 4k · (k 2 + n) · n) time.
Since the r-Swap distance is bounded by the r-Reversal distance (the
corresponding function is rk/2, see Figure 5.1), the above theorem implies
also the existence of an O(rrk (r − 1)rk · 2rk · ((rk)2 + n) · n)-time permissive
parameterized algorithm for LocalTSP(r-Reversal), that is, an algorithm
that returns an improved Hamiltonian cycle whenever there is an improved
Hamiltonian cycle in r-Reversal distance at most k from the given cycle. By
modifying the algorithm from Theorem 5.3.6, we can obtain a strict local
search algorithm for LocalTSP(r-Reversal) with a better running time.
Theorem 5.3.7. LocalTSP(r-Reversal) is fixed-parameter tractable with
respect to the combined parameter (k, r). It is solvable in O(2rk · r2k−1 · (r −
1)k · (k 2 + rk + n) · n) time.
Proof. We modify the algorithm from Theorem 5.3.6. We use the first part
without changing it. That is, the position of the r-reversals are guessed in
the “≤” order in O(r2k−1 · n) time. Again one can show that if the t-th
guessed reversal is ρ(it , jt ), then the (t + 1)-th reversal ρ(it+1 , jt+1 ) must
fulfill it+1 −it ≤ r, as, otherwise, the first t reversals or the last k −t reversals
would yield an improved Hamiltonian cycle themselves.
For the second part we again observe that we do not need to know the
total order of the guessed r-reversals. It is enough to know the order of
reversals ρ(i, j) and ρ(i0 , j 0 ) if {i, i + 1, . . . , j} ∩ {i0 , i0 + 1, . . . , j 0 } 6= ∅ and
(i, j) 6= (i0 , j 0 ). Also observe that for this purpose it suffices to provide the
order on positions where some of the guessed reversals start, that is, on
positions i for which there is a j such that ρ(i, j) is one of the guessed
reversals. Now assume that the orders for all such positions 1 ≤ t0 < t
have been already determined and we want to find the order at position t.
Suppose that there are at reversals ρ(i, j) with i < t ≤ j and bt of them
with t = i < j. Observe that the order of the at reversals is already known
144
from some previous position. We first determine the order of the bt reversals
starting at position t. As there are at most r − 1 types of r-reversals starting
at t, there are at most (r − 1)bt different orders of them. Now it remains to
determine the relative order of the reversals starting at t and those starting
before. With the known orders within these groups, we have less than 2at +bt
such orders. To determine the running time of the algorithm, we multiply
the number of possibilities over all positions. The total number of orders of
the reversals yielding different Hamiltonian cycles is at most
n
Y
(r − 1)bt · 2at +bt = (r − 1)
Pn
t=1
bt
Pn
·2
t=1
at +bt
≤ (r − 1)k · 2rk
t=1
Pn
as the sum t=1 bt of theP
number of reversals starting at some position is
n
at most k, while the sum t=1 at + bt of the number of reversals affecting
the particular position is at most rk for r-reversals.
Once the partial orders at all positions are determined, we check whether
this can be obtained by some total order of the reversals, and find this order in
O(k 2 ) time. Then we apply the reversals in this order in O(kr) time and check
whether we obtain an improved Hamiltonian cycle in linear time. Therefore
the whole algorithm runs in O(2rk · r2k−1 · (r − 1)k · (k 2 + rk + n) · n) time.
Observe that the swap graph of a swap sequence that yields the best
improved Hamiltonian cycle in the local neighborhood does not have to be
connected, and thus Lemma 5.3.5 cannot be extended to this case. However,
we remark that, for LocalTSP(λ) with λ ∈ {r-Swap, r-Reversal}, by
applying a standard dynamic programming approach, the algorithms given
in the proofs of Theorems 5.3.6 & 5.3.7 can be extended such that not
only any improved Hamiltonian cycle is found but also the best improved
Hamiltonian cycle within the local neighborhood.
Further, analyzing the proofs of Theorems 5.3.6 & 5.3.7, one can show
that if there is an improved Hamiltonian cycle in LocalTSP(r-Swap) or
LocalTSP(r-Reversal), then there is also an improved cycle which differs
from the given one only on vertices vi , vi+1 , . . . , vi+rk−1 for some i. Therefore,
one can reduce an input instance to polynomially many instances of the
same problem, each having its size bounded by a polynomial in k and r.
It is enough to replace the part of the cycle between vi+rk−1 and vi by a
length-rk path formed by dummy vertices. Such a self-reduction forms a
Turing kernelization (Definition 2.3.8).
Proposition 5.3.8. LocalTSP(r-Swap) and LocalTSP(r-Reversal) admit a reduction to n Turing kernels, each with at most 2rk vertices, and the
reduction can be computed in linear time.
145
In contrast to Proposition 5.3.8, in the next section we show that LocalTSP(λ) does not admit a polynomial kernel for any distance measure λ
considered in this work, even when restricted to planar graphs.
5.4 Parameterized Complexity of LocalTSP on
Planar Graphs
In this section we investigate the complexity of LocalTSP on planar graphs.
Note that whether LocalTSP(Edge) on planar graphs parameterized by the
locality parameter k is fixed-parameter tractable or not is the central open
question stated by Fellows et al. [Fel+12]. We do not answer this question;
however, we show that on planar graphs LocalTSP(λ) for λ ∈ {Swap,
Edit} is fixed-parameter tractable for parameter k. Before that, we show
that LocalTSP(λ) on planar graphs does not admit a polynomial kernel
for any of the distance measures λ considered in this work.
5.4.1 Non-Existence of a Polynomial-Size Kernel
We prove in detail that LocalTSP(r-Swap) does not admit a polynomial
kernel. As can be seen in Figure 5.1, all distance measures considered in
this work are r-Swap bounded and thus r-Swap can be viewed as the least
powerful distance measure. Thus, a similar argumentation is also valid for
the other distance measures.
To show that LocalTSP(r-Swap) on planar graphs has no polynomial kernel, we first consider a more restricted problem variant, namely
LargeLocalTSP(r-Swap), where it is required that the underlying planar
graph has more than 2rk vertices. We show that LargeLocalTSP(r-Swap)
is NP-hard on planar graphs by a polynomial-time reduction from Weighted
Antimonotone 2-Sat; by exploiting the properties implied by the requirement that there are more than 2rk vertices, we then show that it is
OR-cross-compositional (Definition 2.3.10).
Lemma 5.4.1. LargeLocalTSP(r-Swap) on planar graphs is NP-hard.
Proof. We reduce from the NP-complete Weighted Antimonotone 2-Sat
problem to the LocalTSP(r-Swap) problem and then show how to extend
the construction to LargeLocalTSP(r-Swap).
Weighted Antimonotone 2-Sat
Input: A boolean CNF formula F with two literals in each clause,
both being negative, and a positive integer c.
Question: Is there a satisfying assignment such that at least c variables
are set to True?
146
There is an easy polynomial-time reduction from the NP-complete Independent Set problem to Weighted Antimonotone 2-Sat: For each
vertex there is a corresponding variable and for each edge there is a clause
containing the negative literals of the variables which correspond to the
endpoints of the edge. It is straightforward to argue that the vertices which
correspond to the c variables set to True in a satisfying assignment form an
independent set. The reduction is also valid in the reverse direction.
Let F be a CNF formula with two negative literals per clause which forms
together with a positive integer c an instance of Weighted Antimonotone
2-Sat. We first form a CNF formula F 0 with three negative literals per
clause by adding to F a new so-called dummy variable y such that each
clause of F is extended by the literal ¬y. Furthermore, except for y, for
every variable x in F we add the clause (¬x ∨ y) to F 0 . Observe that the
formulas F and F 0 have a trivial satisfying assignment where all variables
are set to False.
Next, we apply to F 0 a reduction from 3-Sat to Planar Hamiltonian
Cycle due to Garey et al. [GJT76], obtaining a planar graph G such that G
admits a Hamiltonian cycle if and only if F 0 has a satisfying assignment.
Then, the trivial satisfying assignment where all variables are set to False
induces a Hamiltonian cycle in G. Moreover, it follows from the details of the
construction that for every variable there are two edges such that the usage
of these edges in a Hamiltonian cycle specifies whether, in the corresponding
satisfying assignment, the variable has to be set True or False. This means
that there is an edge that is used if and only if the variable is set False and
another edge that is used if and only if the variable is set True. We briefly
refer to them by the False-edge and True-edge, respectively.
In order to form a LocalTSP(r-Swap) instance, we assign weight one
to every False-edge. Hence, if F 0 contains n variables, then the satisfying
assignment where all variables are set to False has weight n. Finally, assign
weight c to the True-edge of y, whereas all remaining edges have
weight
zero. Every permutation of t elements can be sorted by at most 2t 2-swaps.
Hence, denoting the Hamiltonian
cycle through G that uses all False-edges
by id and setting k = |V (G)|
allows
to choose any permutation as a solution
2
of the LocalTSP(r-Swap) instance (G, ω, k).
We next show the correctness of the construction above, meaning that
there is a satisfying assignment for F with at least c variables set to True
if and only if there is a Hamiltonian cycle through G with weight strictly
less than n. First, assume that there is a satisfying assignment for F with at
least c variables set to True. It is clear that extending this assignment by
setting y = True we also get a satisfying assignment for F 0 . Moreover, since
in the corresponding Hamiltonian cycle through G at least c zero-weight
True-edges are used, and since the weight-c True-edge for y is used instead
147
of the weight-1 False-edge, the weight of the Hamiltonian cycle is at most
n + c − 1 − c < n.
For the reverse direction, assume that there is a Hamiltonian cycle with
weight strictly less than n and consider the corresponding satisfying assignment for F 0 . Then, there is at least one variable, say x, that is set to True.
Since the clause (¬x ∨ y) ∈ F 0 has to be satisfied, it follows that y is also
set to True. Hence, removing y from the assignment results in a satisfying
assignment for F and since the True-edge of y has weight c but the weight
of the cycle is less than n, in total there have to be at least c other variables
set to True.
The presented argumentation shows that LocalTSP(r-Swap) is NP-hard
on planar graphs. Moreover, it is clear from the details of the construction
by Garey et al. [GJT76] that there is an edge (for example, the edge from
the last clause-gadget to the first variable-gadget) that has to be used in any
Hamiltonian cycle. Thus, subdividing this edge an appropriate number of
times, we also get an equivalent LargeLocalTSP(r-Swap) instance.
In order to apply the cross-composition framework (Definition 2.3.10),
it remains to show that LargeLocalTSP(r-Swap) on planar graphs is
OR-cross-compositional. To prove this, we first need the following easy
observation.
Observation 5.4.2. For a LargeLocalTSP(r-Swap) instance (G, ω, k),
let π ∈ Sn be an improved Hamiltonian cycle that can be obtained by a
sequence S of at most k r-swaps. Then, there is an edge e = {vt , vt+1 } ∈
E(π) ∩ E(id) such that the vertices vt and vt+1 are unaffected and no swap
goes over these vertices. Formally, this means that for all σ(i, j) ∈ S it holds
that neither t nor t + 1 is contained in {i, i + 1, . . . , j − 1, j}.
Proof. A LargeLocalTSP(r-Swap) instance fulfills n > 2kr. Then, since
performing k r-swaps cannot affect more than 2k vertices, it follows that there
are some r consecutive vertices that are unaffected. The second statement
follows because one r-swap cannot “span” more than r − 1 vertices and thus
no swap affects or goes over these vertices.
Lemma 5.4.3. LargeLocalTSP(r-Swap) on planar graphs is OR-crosscompositional.
Proof. We provide an OR-cross-composition from LargeLocalTSP(r-Swap)
to itself. For some t ∈ N, let (G1 , ω1 , k), (G2 , ω2 , k), . . . , (Gt , ωt , k) be instances of LargeLocalTSP(r-Swap). For each graph Gi with 1 ≤ i ≤ t
we introduce several copies of Gi , and in each copy we choose a start and
an end vertex. Then, a graph G with weight function ω is composed by
arranging the copies in an arbitrary order and connecting the end vertex
148
of a copy by a zero-weight edge to the start vertex of the subsequent copy.
Finally, the end vertex of the last copy is connected by a zero-weight edge
to the start vertex of the first copy. We describe a Hamiltonian path from
the start to the end vertex of each copy. All these paths together with the
zero-weight edges between the copies form the Hamiltonian cycle id and
(G, ω, k) forms the composed instance. Furthermore, since the start and the
end vertex of each copy are connected by an edge, and, therefore, there is an
embedding of the copy with both vertices on the boundary of the outer face,
graph G is planar.
In order to form the composed graph G, for 1 ≤ i ≤ t and ni = |V (Gi )|
let v1i , v2i , . . . , vni i , v1i be the given Hamiltonian cycle in Gi . For 1 ≤ j < ni ,
i
we add one copy of Gi with the vertex vj+1
being the start vertex and vji
being the end vertex. The given Hamiltonian cycle in Gi then induces a
i
i
i
Hamiltonian path vj+1
, vj+2
, . . . , vni i , v1i , . . . , vji from vj+1
to vji in this copy.
We complete the construction by adding a copy of Gi where v1i is the start
and vni i is the end vertex.
In the following we prove the correctness of the reduction, that is, the
composed instance (G, ω, k) is a yes-instance of LargeLocalTSP(r-Swap)
if and only if there is a yes-instance (Gi , ωi , k) with 1 ≤ i ≤ t. First,
suppose that there is an improved Hamiltonian cycle for G which performs
at most k swaps. By our construction it is obvious that any Hamiltonian
cycle through G enters a copy at its start vertex and leaves it at its end
vertex. Thus, there is at least one copy where the improved Hamiltonian
cycle for G implies an improved Hamiltonian path from the start to the end
vertex. As the improved Hamiltonian path does not use the edge between the
end vertex and the start vertex of the particular copy, by adding this edge
we obviously get an improved Hamiltonian cycle in the corresponding graph,
which is in r-Swap distance at most k from the given Hamiltonian cycle.
For the reverse direction, suppose that there is an improved Hamiltonian
cycle for the graph Gi . By Observation 5.4.2 there is at least one “preserved”
i
edge {vji , vj+1
} or {vni i , v1i }, where no swap goes over its endpoints. It is
i
clear that the same swaps are also r-swaps in the copy of Gi with vj+1
(or
i
i
i
v1 ) as a start vertex and vj (or vni ) as the end vertex. Hence all swaps
which were performed to Gi can be performed to this copy, resulting in an
improved Hamiltonian cycle for G.
By Lemma 5.4.3 and Lemma 5.4.1 there is a OR-cross-composition from
the NP-hard LargeLocalTSP(r-Swap) problem to itself and thus by
Theorem 2.3.11 it follows that LargeLocalTSP(r-Swap) does not admit a
polynomial kernel, unless NP ⊆ coNP/poly. Thus, the next theorem follows.
149
Theorem 5.4.4. Unless NP ⊆ coNP/poly, LocalTSP(r-Swap) on planar
graphs does not admit a polynomial kernel with respect to the parameter k
for any r ≥ 2.
Following exactly the same argumentation as for LocalTSP(r-Swap),
we state that one can show that on planar graphs LocalTSP(λ) for λ ∈ {rReversal (for any r ≥ 2), Swap, Edit, Reversal, Max-Shift, Edge} does
not admit a polynomial kernel with respect to parameter k, unless NP ⊆
coNP/poly.
Corollary 5.4.5. Unless NP ⊆ coNP/poly, LocalTSP(λ) for λ ∈ {r-Reversal,
Swap, Edit, Reversal, Max-Shift, Edge} on planar graphs does not admit a
polynomial kernel with respect to the parameter k for any r ≥ 2.
5.4.2 Fixed-Parameter Tractability of LocalTSP(Edit)
and LocalTSP(Swap)
LocalTSP(Edit) and LocalTSP(Swap) on planar graphs are unlikely to
allow for polynomial kernels; however, they admit a permissive parameterized
algorithm. In the following we argue for LocalTSP(Swap); the result for
the Edit distance can be obtained along the same lines. The proof relies on
the following two lemmas.
Lemma 5.4.6. If a LocalTSP(Swap) instance with parameter k admits
an improved Hamiltonian cycle, then it also admits an improved Hamiltonian cycle which differs from the given one only within the distance-3k
neighborhood around some vertex.
Proof. Due to Lemma 5.3.5 from Section 5.3.2, it suffices to prove the
statement only for improved Hamiltonian cycles obtained by a sequence of
swaps where the corresponding swap graph is connected. Consider the set A
of all vertices affected by swaps and their neighbors in the given Hamiltonian
cycle Q. Since there are at most k swaps we have |A| ≤ 6k. Clearly, the
improved Hamiltonian cycle R coincides with Q outside A.
Now we consider the connected components of G[A]. Furthermore, we also
consider some maximal path P of R such that P is formed only by vertices
of A. Obviously, P contains vertices of only one component of G[A]; call this
component C. Let x, y be the two neighbors of P on R. Since the part of Q
outside of A is preserved, the path P 0 of Q between x and y also contains
only vertices of C. Moreover, since x and y are not affected by any swap and,
therefore, there must be the same number of vertices between them in Q
and in R, path P 0 has the same length as P . By repeating the argument,
one can show that C can be partitioned into such paths of Q and R in a
150
one-to-one correspondence. Therefore, one can obtain R in C from Q only
using swaps within C.
Due to the above argument, it suffices to consider the case that each swap
is within one component of G[A]. Observe that swaps in different components
of G[A] are not adjacent in the swap graph. Therefore, since we assumed the
swap graph to be connected, G[A] has only one component, that is, G[A] is
connected. Finally, in a connected graph with at most 6k vertices there is at
least one vertex that has distance at most 3k to all others.
The following lemma shows that, regardless of the distance measure, on
planar graphs it is fixed-parameter tractable to find the best improved Hamiltonian cycle that differs from the given one only within the neighborhood of
one specific vertex.
Lemma 5.4.7. For an instance of LocalTSP on planar graphs and a
vertex v one can find in O(2O(k) · n + n3 ) time the best Hamiltonian cycle
among those differing from the given one only within distance k from v.
Proof. Start by deleting the edges having an endpoint of distance greater
than k from v which are not part of the given Hamiltonian cycle (they cannot
be used by the new Hamiltonian cycle anyway). Now contract the paths of
the given Hamiltonian cycle formed by vertices in distance more than k from v
into one vertex (to avoid duplicate edges). Then the whole graph is still planar
and has diameter at most 2k + 2 and, therefore, by a result of Robertson
and Seymour [RS86] (see also [Bod98]) has treewidth at most 6k + 6. Thus
it has branch-width at most 6k + 6 [RS91]. Dorn et al. [Dor+10] showed that
TSP on planar graphs (referred to as Planar Hamiltonian Cycle) with
branch-width l can be solved in O(23.292l · l · n + n3 ) time. To modify their
algorithm for our problem it is enough to force their algorithm to preserve
the edges of paths that represent the parts of the given Hamiltonian cycle in
distance more than k from vertex v. This is easy, as Dorn et al. [Dor+10]
basically consider the solution to be the set of edges of the Hamiltonian
cycle, and the branch-decomposition based dynamic programing actually
starts with individual edges. It is enough to fill the tables so that the only
solution on a required edge is to take this edge.
Theorem 5.4.8. There exists a permissive parameterized algorithm for
LocalTSP(λ) on planar graphs with respect to k for λ ∈ {Swap, Edit}.
Proof. We prove the theorem only for the Swap distance, the result for the
Edit distance can be obtained along the same lines. Assume that there is
an improved Hamiltonian cycle in Swap distance at most k from the given
Hamiltonian cycle. By Lemma 5.4.6 we know that in this case there is an
improved Hamiltonian cycle, differing from the given one only within distance
151
at most 3k from some vertex v. We can find such a Hamiltonian cycle or a
Hamiltonian cycle that is at least as good, by applying the algorithm from
Lemma 5.4.7 on the 3k-neighborhood of each vertex. The O(n·(2O(k) ·n+n3 ))
running time follows.
Following the same approach as Fellows et al. [Fel+12], Theorem 5.4.8 can
be easily generalized to any class of graphs with bounded local treewidth.
As Lemma 5.4.6 does not assume anything about the graph, we only have
to modify Lemma 5.4.7. The lemma is true in any class of graphs with
bounded local treewidth, but the corresponding running time depends on
the respective class.
5.5 Conclusion and Open Questions
We left open the central open problem posed by Fellows et al. [Fel+12]
whether LocalTSP(Edge) restricted to planar graphs is fixed-parameter
tractable. However, we indicated (see Section 5.2) that a permissive parameterized algorithm for LocalTSP(Edge) implies a permissive parameterized
algorithm for LocalTSP(Reversal) and vice versa. Thus, the questions
whether the problems are fixed-parameter tractable, are equivalent and this
might help to shed new light on this question. To this end, it might be
beneficial to explore the connections of LocalTSP(Reversal) to the topic
of Sorting by Reversals as studied in bioinformatics [Cap97]. Moreover,
it might be worthwhile to explore whether there are strict algorithms for
LocalTSP(λ) for λ ∈ {Swap, Edit} on planar graphs.
Assuming the ETH (Hypothesis 2.3.15), we showed that there is no
O(no(k/ log k) )-time algorithm for LocalTSP(λ) for λ ∈ {Swap, Edit, Reversal, Edge}. Is there also a matching upper bound or can the lower
bound still be improved to match the best algorithms known so far running
in O(nk ) time?
Finally, our investigations might also be extended by moving from local
neighborhoods for TSP to so-called exponential (but structured) neighborhoods as undertaken already in a non-parameterized setting [DW00, GYZ02].
152
6 Parameterizing Vector
Explanation by Properties of
Numeric Vectors
We extend previous studies on NP-complete problems dealing with the
decomposition of nonnegative integer vectors into sums of few homogeneous
segments. These problems are motivated by radiation therapy and database
applications. In one problem, called Vector Positive Explanation, the
segments may have only positive integer entries, in the other problem, called
Vector Explanation, the segments may have arbitrary integer entries.
Considering several structural parameterizations such as the maximum
vector entry γ and maximum difference between consecutive vector entries δ,
we obtain a refined picture of the computational (in-)tractability of these
problems. For example, we show that Vector Explanation is fixedparameter tractable with respect to δ, and that, unless NP ⊆ coNP/poly,
there is no polynomial kernelization for Vector Positive Explanation
with respect to the parameter γ. We also identify relevant special cases where
Vector Positive Explanation is algorithmically harder than Vector
Explanation.
6.1 Introduction
We investigate two variants of a “mathematically fundamental” [Ban+11],
NP-complete combinatorial problem motivated by cancer radiation therapy
planning [Ehr+10] and database and data warehousing applications [Aga+07,
Kar+11]. Let N = {0, 1, 2, . . .}. We examine the Vector Explanation
and Vector Positive Explanation problem.
Vector (Positive) Explanation
Input: A vector A ∈ Nn and an integer k.
Question: Can A be explained by at most k (positive) segments?
Herein, a segment is a vector in {0, a}n for some a ∈ Z \ {0} where
all a-entries occur consecutively, and a segment is positive if a is positive.
153
-1
1
4
[ 4
-4
3
3
4 ]
A4
3
3
4
4
43
34
4
Figure 6.1: Illustration of the geometric interpretation of an input vector A =
(4, 3, 3, 4) (left hand side), an explanation of it using only positive segments (middle),
and an explanation with one negative segment (dotted pattern) on the right-hand
side. Vector A is represented by a tower of blocks where each position i on the
x-axis has A[i] many blocks. Each segment I ∈ {0, a}n is represented by a height-a
rectangle starting and ending in the corresponding first and last a-entry of I
(their different positions on the y-axis are only to draw them in a non-intersecting
fashion). Having a set of segments, they explain A if for each i the sum of the
heights of the rectangles intersecting a position i on the x-axis is A[i].
An explanation is a set of segments that component-wisely sum up to
the input vector.
For example, in case of Vector Explanation (VE for short) the vector
(4, 3, 3, 4) can be explained by the segments (4, 4, 4, 4) and (0, −1, −1, 0),
and in case of Vector Positive Explanation (VPE for short) it can
be explained by (3, 3, 3, 3), (1, 0, 0, 0), and (0, 0, 0, 1). Observe that, when
restricted to use only positive segments as in VPE one needs at least three
segments to explain (4, 3, 3, 4). We will prove that the “gap” between the
sizes of minimum explanations for both problem variants is at most two. In
addition, both problems have a simple well-known geometric interpretation
(see Figure 6.1).
VE occurs in the database context and VPE occurs in the radiation therapy context. Motivated by previous work providing polynomial-time solvable
special cases [Aga+07, Ban+11], polynomial-time approximation [Bie+11,
LSY07], and fixed-parameter tractability results [Bie+13, COO12] (approximation and parameterized algorithms both exploit problem-specific structural parameters), we head for a systematic classification of tractable and
intractable cases of structural parameterizations. In this way we follow the
second approach the identify tractable cases of NP-hard problems (Chapter 1). Interestingly, by this methodology we found parameters for which
the parameterized complexity of both problem variants differ.
154
6.1.1 Related Work
Agarwal et al. [Aga+07] studied a polynomial-time solvable variant (“treeordered”) of VE relevant in data warehousing. Karloff et al. [Kar+11]
initiated a study of (special cases of) the two-dimensional (“matrix”) case
of VE and provided NP-completeness results as well as polynomial-time
constant-factor approximations. Parameterized complexity aspects of VE
and its two-dimensional variant seem unstudied so far.
The literature on VPE is richer. For a description of the motivation
from radiation therapy refer to the survey due to Ehrgott et al. [Ehr+10].
Concerning the computational complexity, VPE is known to be strongly
NP-complete [Baa+05] and APX-hard [Ban+11]. A significant amount of
work has been done to achieve a factor- 24
13 polynomial-time approximation
algorithm for minimizing the number of segments which improve on the
straightforward factor of two [Ban+11] (see also Biedl et al. [Bie+11]).
Improving a previous fixed-parameter tractability result for the parameter “maximum value γ of a vector entry” by Cambazard et al. [COO12],
Biedl
et al. [Bie+13] developed a parameterized algorithm solving VPE in
√
2O( γ) · γn time with n being the number of entries in the input vector.
Moreover, the parameter “maximum difference δ between two consecutive
vector entries” has been exploited for developing polynomial-time approximation algorithms [Bie+11, LSY07]. More precisely, Biedl et al. [Bie+11]
provide polynomial-time approximation algorithms with approximation factors of (roughly) 32 (1 + log3 γ) and 24
13 log δ and also present an experimental
evaluation of their algorithms. Finally, we remark that most of the previous
studies also looked at the two-dimensional (“matrix”) case, whereas we focus
on the one-dimensional (“vector”) case.
6.1.2 Our Contribution
We observe that the combinatorial structure of the considered problems is extremely rich, opening the way to a more thorough study of the computational
complexity landscape under the perspective of structural parameterization.
We work through the corresponding parameter space following the strategy to look for stronger parameterizations (Definition 2.3.14), for example,
parameters which are motivated by the “distance from triviality” perspective [GHN04]. In this way we take a closer look at parameterization aspects
that help in better understanding and exploiting problem-specific properties.
To start with, note that previous work [Bie+13, COO12], motivated
by the application in radiation therapy, studied the parameterization by
the maximum vector entry γ. They showed fixed-parameter tractability
for VPE parameterized by γ, which we complement by showing the non-
155
-1
-1
2
2
-1
-1
2
2
-4
-4
1
1
1
1
A= [ 1
2
4
4
4
3
4
3
5
0
4 ]
A =[ 1
2
4
3
5
4 ]
stretch it
1
2
4
4
1
2
4
4
4
3
Figure 6.2: Illustration of Reduction Rule 6.1.1. The input vector A (top left) has
consecutive entries which are equal. In the bottom left part there is a (partial)
explanation of A containing a segment ending between A[3] = A[4]. By “stretching”
these kinds of segments one can get an explanation in which no segment ends or
starts within equal consecutive entries (bottom right part). It is straightforward
to modify such an explanation to get an explanation for the input vector A0 (top
right) which results from A by exhaustively applying Reduction Rule 6.1.1.
existence (using the cross-composition framework, see Definition 2.3.10)
of a corresponding polynomial kernel. Using an integer linear program
formulation, we also show fixed-parameter tractability for VE parameterized
by γ. Moreover, for the perhaps most obvious parameter, the number k of
explaining segments, we show fixed-parameter tractability for both problems.
Further Parameters under Study. Before describing further results, in
order to define the corresponding parameters, we need to introduce an
already known data reduction rule [Ban+11].
Reduction Rule 6.1.1. If the input vector A has two consecutive equal
entries, then remove one of them.
The correctness of Reduction Rule 6.1.1 is obvious, as there is always
a minimum-size explanation such that for each segment S it holds that
S[i] = S[i + 1] in case of A[i] = A[i + 1] (see Figure 6.2 for an illustration).
For notational convenience, we use A[0] = A[n + 1] = 0 and thus in case that
156
A[0] = A[1] = 0 or A[n] = A[n + 1] = 0 we also apply Reduction Rule 6.1.1
to them. We emphasize that we consider neither A[0] nor A[n + 1] as part
of the vector A ∈ Nn . The following lemma states the central consequence
of an exhaustive application of Reduction Rule 6.1.1.
Lemma 6.1.2. Any instance of Vector Positive Explanation or Vector Explanation can be reduced in O(n) time to an equivalent one with
at most (2k − 1) entries.
It is easy to observe that Reduction Rule 6.1.1 can be applied in O(n) time
to an input vector A ∈ Nn . For any input vector A ∈ Nn that is reduced
with respect to Reduction Rule 6.1.1 it holds for each position 0 ≤ i ≤ n that
there is at least one segment S in each explanation such that S[i] 6= S[i + 1],
implying that if n + 1 > 2k, then the instance is a trivial no-instance. On
the other hand, k ≥ n would allow to use one segment for each input vector
entry and thus the instance would be a trivial yes-instance. Hence, we may
assume throughout the rest of this work that A[i] 6= A[i + 1] for all 0 ≤ i ≤ n
and k < n < 2k. Observe further that Lemma 6.1.2 implies that taking a
segment for each input vector entry is a trivial factor-two approximation for
VPE as well as for VE.
We now have the ingredients to provide a formal definition of all parameters
considered in this work.
Definition 6.1.3. For an input vector A ∈ Nn define:
• maximum difference between consecutive entries δ = max0≤i≤n |A[i] −
A[i + 1]|;
• number of peaks p where a position 1 ≤ i ≤ n is a peak if A[i − 1] <
A[i] > A[i + 1];
• maximum value γ = max1≤i≤n A[i];
• number k of allowed segments in an explanation;
• “distance from triviality”-parameters n − k and φ = 2k − n;
• maximum segment length ξ (number of non-zero entries) in an explanation;
• maximum number o of segments in an explanation which have a nonzero entry at a particular vector entry.
Note that, since we may assume by the above discussion that k < n <
2k, the parameters n − k and φ are well defined. Indeed, both can be
interpreted as “distance from triviality” parameterizations [GHN04, Nie10].
157
Table 6.1: An overview of previous and new results.
Parameters
Vector Explanation
γ)
· γn [Bie+13]
no poly. kernel (Thm. 6.3.4)
√
O(nδ · eπ 2δ/3 ) (Thm. 6.3.3)
FPT (Cor. 6.3.2)
max. difference δ of
consecutive entries
(# of peaks p, δ)
FPT
(Thm. 6.3.1)
k O(k) + nO(1) (Thm. 6.4.1)
number k of segments
(2k − 1)-entry kernel (Thm. 6.4.1)
φO(φ) + nO(1) (Thm. 6.4.6(ii)) k O(φ) + nO(1) (Thm. 6.4.6(i))
3φ-entry kernel (Thm. 6.4.6(ii))
n−k
W[1]-hard (Thm. 6.4.7)
NP-complete for (n − k) = 1 (Thm. 6.4.8)
ξ ≥ 3 : NP-complete (Thm. 6.4.8)
max. segment
length ξ
max. number o of
overlapping segments
√
2O(
max. value γ
φ = 2k − n
Vector Pos. Explanation
ξ ≤ 2 : O(n2 ) (Thm. 6.4.9)
o = 1: trivial
o = 2 (and ξ = 3 and n − k = 1): NP-complete (Thm. 6.4.8)
Furthermore, observe that the last two parameters (max. segment length and
max. number non-zero entries in an explanation) are, in difference to the other
parameters, rather structural parameterizations of a solution/explanation
than of the input.
Table 6.1 summarizes our and previous results with respect to the parameters defined in Definition 6.1.3. We show that, somewhat surprisingly,
VE and VPE are already NP-complete for n − k = 1. Furthermore, we
will show that instances with k = bn/2c + 1 are polynomial-time solvable,
motivating the study of parameter φ. Interestingly, while we show that
VPE is W[1]-hard for parameter φ, we show that VE is fixed-parameter
tractable for φ. Finally, we show NP-completeness for VE and VPE when
ξ = 3 and o = 2.
6.1.3 Organization
In Section 6.2, we present a number of useful combinatorial properties of
vector explanation problems which may be of independent interest and which
are used throughout our work. In Section 6.3, we study the “smoothness of
input vector-parameters” γ, δ, and p. In Section 6.4, we present results for
further parameters as discussed above, and we conclude in Section 6.5 with
some challenges for future research.
158
6.2 Further Notation and Combinatorial
Properties
We say that a segment I ∈ {0, a}n for some a ∈ Z \ {0} is of weight a and
it starts at position l and ends at positions r if I[i] = a for all 1 ≤ l ≤ i <
r ≤ n and all other entries are zero. We will briefly write [l, r] for such a
segment and we say that it covers position i whenever l ≤ i < r.1 Because
this notation suppresses the weight of the segment, we will associate a weight
function ω : I → Z with a set I of segments that relates each segment to its
weight. A set I of segments with a corresponding weight function ω forms
an explanation for A ∈ Nn if for each 1 ≤ i ≤ n the total sum of weights
of all segments covering position i is equal to A[i]. We also say that (I, ω)
explains A and refer to |I| as solution size. Segments with positive weight
are called positive segments, those with negative weight are called negative
segments. In case of VPE, we only allow positive segments.
Since we assume that in a preprocessing phase Reduction Rule 6.1.1 is
exhaustively applied, without loss of generality it holds that A[i] 6= A[i + 1]
for all 0 ≤ i ≤ n. It will turn out that the difference between consecutive
entries in A is an important quantity.
Definition 6.2.1 (Tick Vector). For an input vector A ∈ Nn we define the
tick vector T ∈ Nn+1 to be T [i] = A[i] − A[i − 1] for all 1 ≤ i ≤ n + 1.
We call A the input vector corresponding to the tick vector T . A position
1 ≤ i ≤ n + 1 is called an uptick if T [i] > 0 and otherwise it is called a
downtick. The size of the corresponding up- or downtick is |T [i]|.
Note that given a tick vector
Pi T the corresponding input vector A is
uniquely determined as A[i] = j=1 T [j]. Thus, we will call an explanation
for A also an explanation for its tick vector T . Observe that the parameter
maximum difference δ between consecutive entries is the maximum absolute
value in T .
We next define a structure for an explanation and subsequently prove that
there is always a minimum-size explanation of this structure.
Definition 6.2.2 (Regular Explanations). An explanation is called regular
if each positive segment starts at an uptick and ends at a downtick, and each
negative segment starts at a downtick and ends at an uptick.
By the following theorem we can assume that each explanation is regular.
For VPE it corresponds to Bansal et al. [Ban+11, Lemma 1].
1 Note
that [l, r] does not cover position r, but it covers position l.
159
(i)
(iii)
(ii)
−α
α
β
α+β
β
α+β
Figure 6.3: Illustration of the three configurations where two segments are called
messy overlapping. That is, (i) both segments, one with weight α and one with
weight β, start at the same position. In configuration (ii) the first segment ends at
the same position where the second starts. Configuration (iii) is when both end
at the same position. Given two messy overlapping segments in an explanation
one can transform them into any of the other configurations as depicted while
still having an explanation. Transforming two segments of equal weight from
configuration (ii) to any of (i) or (iii) would require to insert a zero-weight segment
which will be omitted.
Theorem 6.2.3. Let (I, ω) be a size-k explanation of an input vector A.
There is a regular size-k explanation (I 0 , ω 0 ) for A such that (I 0 , ω 0 ) contains
only positive segments if (I, ω) does so.
Proof. Let A be an input vector and let (I, ω) be a non-regular explanation
of A. We say that a segment I ∈ I has a wrong start if I is positive and
starts at a downtick or if I is negative and starts at an uptick. Otherwise,
we say the start is correct. We define wrong and correct ends analogously.
Correspondingly, the start (end) of a wrong start (wrong end) segment is
called wrong start (wrong end) position.
Let A = (A[n], A[n − 1], . . . , A[1]) be the vector formed by reversing A
and let I be the set of segments formed by reversing each segment in I.
Clearly, (I, ω) is an explanation for A. Hence, we may assume that in A
there is a wrong start segment, as we otherwise consider A. We will provide
a restructuring procedure whose application to I does not decrease the
smallest (leftmost) wrong start position, the sum of the absolute weights
of the wrong start segments starting at the smallest wrong start position
of I strictly decreases, and it does not increase the number of wrong end
positions. Thus by iteratively applying this restructuring one can “remove”
from left to right all wrong start segments. Then the reversal vector A does
not have any wrong end segments and thus by applying the same procedure
again to A one removes all wrong end segments in A without introducing
any new wrong start segments.
We now describe the restructuring procedure. Let I = [li , ri ] ∈ I be
a wrong start segment starting at the smallest wrong start position li .
Since I is a wrong start segment there is a segment J = [lj , rj ] with lj ≤ li
160
such that either the sign of the weights of I and J are equal and J ends
at li (Case 1), or J has an opposite weight sign and starts at li (Case 2).
Clearly, Case 2 occurs only if explanation I contains negative segments.
In Case 1, our restructuring procedure only introduces segments of the same
weight sign as I. (This ensures that the restructured explanation contains
negative segments only if (I, ω) does so.) In either case, I and J are called
to be in a messy overlapping configuration, that is, either they start at
the same position (configuration (i)) or one segment ends where the other
starts (configuration (ii)). We will also call the configuration where two
segments end at the same position messy overlapping (configuration (iii)).
See Figure 6.3 how to transform the configurations into each other while
preserving an explanation.
Case 1: J ends at li , and ω(J) and ω(I) have equal sign.
Thus J = [lj , li ] and I, J are in configuration (ii), implying that ω(J) =
α + β and ω(I) = β in the terminology of Figure 6.3. If α > 0, then
transform I, J into configuration (i) and otherwise transform them into
configuration (iii). In both cases this decreases the absolute weight sum of
wrong start segments starting at li and does not decrease the smallest wrong
start position as J has a correct start. Additionally, observe that we neither
introduced new wrong end positions nor, if ω(I) and ω(J) both are positive,
created negative segments.
Case 2: J starts at li and ω(J) and ω(I) have different signs.
Thus, J = [li , rj ] and I, J are in configuration (i), implying that, when using
the terminology of Figure 6.3, α and β have different signs. If |β| ≥ |α|,
then we transform I, J into configuration (iii). Note that as α and β have
different signs this strictly decreases the absolute weight sum of wrong start
segments at position li . Furthermore, as |β| ≥ |α| and thus the signs of α + β
and −α are the same as of β, it neither creates new wrong end positions nor
decreases the smallest wrong start position. If |α| > |β|, then transform I, J
into configuration (ii). Again, since α and β have different signs this strictly
decreases the absolute weight sum of wrong start segments starting in li and
because |α| > |β|, the two segments with weight α + β and β have the same
sign as α and β, respectively. Hence, this transformation does not introduce
new wrong end positions.
Remark. Note that Theorem 6.2.3 implies containment in NP for VE as
it upper-bounds the segment weights in an explanation in the numbers
occurring in the instance. In case of VPE this directly follows from the
problem definition.
The following corollary summarizes the consequences of Theorem 6.2.3.
To state them, we introduce the following terminology.
161
Definition 6.2.4 (Single-Peaked Vectors). An input vector is single-peaked
if it contains only one peak. A single-peaked instance is an instance with a
single-peaked vector.
Corollary 6.2.5. (i) For any Vector Positive Explanation or Vector Explanation instance there is a minimum-size explanation such
that there is only one segment that covers the first position and it is
positive and ends at a downtick. Symmetrically, there is a minimumsize explanation such that there is only one segment that covers the
last position and it is positive and starts at an uptick.
(ii) If a Vector Explanation instance (A, k) is single-peaked, then (A, k)
is an equivalent Vector Positive Explanation instance.
Proof. Corollary 6.2.5(i): Since position 1 is an uptick and n + 1 a downtick,
by Theorem 6.2.3 it directly follows that in a regular explanation all segments
covering the first or last position are positive and thus start in upticks and
end in downticks. Moreover, if there are two positive segments covering the
first position, then they are messy overlapping as they are in configuration (i)
(Figure 6.3). Hence, transforming them into configuration (ii) results in an
explanation where one segment less covers the first position. Analogously, two
segments covering the last position can be transformed from configuration (iii)
to configuration (ii).
Corollary 6.2.5(ii): By Theorem 6.2.3 if there is any size-k explanation,
then there is also a regular size-k explanation which starts negative segments
in downticks and ends them in upticks. However, in single-peaked instances
all upticks precede the first downtick.
The following theorem states that for VE one can arbitrarily reorder a tick
vector without changing the solution size for the corresponding input vectors.
Theorem 6.2.6. Let T ∈ Nn+1 be an arbitrary tick vector and let T 0 ∈ Nn+1
be a tick vector that results from T by arbitrarily reordering the entries in T .
For Vector Explanation it holds that there is a size-k explanation for T
if and only if there is a size-k explanation for T 0 .
Proof. We prove Theorem 6.2.6 for two tick vectors T and T 0 where, for
some i, T 0 [i] = T [i + 1], T 0 [i+1] = T [i], and T [j] = T 0 [j] for all other entries j.
It is clear that one can arbitrarily reorder T by applying these “flips” to
consecutive entries. Let A be the input vector corresponding to T and let A0
be the input vector corresponding to T 0 . It follows that A0 [j] = A[j] for every
j 6= i and A0 [i] = A[i − 1] + A[i + 1] − A[i]. For any k, we prove that (A0 , k)
is a yes-instance if and only if (A, k) is a yes-instance. However, as “flipping”
T 0 [i] and T 0 [i + 1] in T 0 results in T , the equivalence is symmetric and it is
thus sufficient to prove that if (A, k) is a yes-instance, then so is (A0 , k).
162
Let (I, ω) be an explanation for A. We construct (I 0 , ω 0 ) by replacing
some segments in I. The general idea is that if a segment started or ended
at position i, then it is modified such that it starts or ends at i + 1 and
vice versa. The only exception are the segments which start at i and end
at i + 1, for which we swap the endpoints and negate the weight. Formally,
I 0 is defined as follows:
I 0 = I00 ∪0 I10 ∪ I20 ∪ I30 ∪ I40 ∪ I50 ∪ I60 , where
I00 = {[a, b] ∈ I | a, b < i ∨ a, b > i + 1},
I10 = {[a, b] ∈ I | a < i ∧ b > i + 1},
I20 = {[a, i + 1] | [a, i] ∈ I},
I30 = {[a, i] | [a, i + 1] ∈ I ∧ a < i},
I40 = {[i + 1, b] | [i, b] ∈ I ∧ b > i + 1},
I50 = {[i, b] | [i + 1, b] ∈ I},
I60 = {[i, i + 1]} ∩ I.
Let ω 0 ([i, i+1]) = −ω([i, i+1]) if [i, i+1] ∈ I, and for the other segments of I 0
set the weight ω 0 to be equal to the weight of the corresponding segment in I.
Obviously, |I 0 | = |I| and, hence, it remains to show that (I 0 , ω 0 ) explains A0 . As a segment of I 0 covers a position j 6= i if and only if the
corresponding segment in I of the same weight covers j, it is clear that
0
(I 0 , w0 ) explains every position
PA [j] =0 A[j] with j 6= i. To prove that it also
explains position i, let sx = I∈Ix0 ω (I) for all x ∈ {1, . . . , 6}. Since (I, w)
explains A and the weight of the segments (except those potentially in I60 )
are equal, it holds that
A[i − 1] = s1 + s2 + s3 ,
A[i] = s1 + s3 + s4 − s6 , and
A[i + 1] = s1 + s4 + s5 .
The sum of the weights of segments covering A0 [i] is s1 + s2 + s5 + s6 and
thus together with A0 [i] = A[i − 1] + A[i + 1] − A[i], the equations above
prove that (I 0 , w0 ) also explains A0 [i].
The following corollary summarizes combinatorial properties of VE which
can be directly deduced from Theorem 6.2.6 as it allows to arbitrarily order
the entries of a tick vector. They are used throughout this work and may be
of independent interest for future studies.
Corollary 6.2.7. Let (A, k) be an instance of Vector Explanation.
Then, the following holds.
163
(i) The instance (A, k) can be transformed in O(n) time to an equivalent
single-peaked Vector Explanation-instance (A0 , k) such that the
maximum difference between consecutive entries is the same in A
and A0 .
(ii) The instance (A, k) can be transformed in O(n) time to an equivalent
Vector Explanation-instance (A0 , k) where the maximum value
in A0 is less than two times the maximum difference between consecutive
entries in A.
Proof. Corollary 6.2.7(i): By Theorem 6.2.6, reordering the entries of the
tick vector of A such that all upticks precede all downticks results in an
equivalent instance. Clearly, this can be done in O(n) time.
Corollary 6.2.7(ii): Let δ be the maximum difference between consecutive
entries in A, or equivalently, the maximum absolute value in the tick vector T
of A. Start creating a reordering T 0 of T by assigning T [1] to be an arbitrary
uptick from T . Next, whenever T 0 [1], . . . , T 0 [i − 1] are already assigned, if
Pi−1
j=1 T [j] < δ and there is an uptick in T that is not yet assigned to one
of T 0 [1], . . . , T 0 [i − 1], then assign T [i] to be this uptick. Otherwise set it to
one downtick of T that is not yet assigned. Clearly, a partition of T ’s entries
in up- and downticks can be computed in O(n) time and using it one can
easily do the above assignment.
6.3 Parameterization by Input Smoothness
In this section, we examine how the computational complexity of VE
and VPE is influenced by parameters that measure how “smooth” the input
vector A ∈ Nn is. We assume that A is reduced with respect to Reduction
Rule 6.1.1 and thus all consecutive positions in A have different values. We
consider the following three measurements:
• the maximum difference δ between two consecutive values in A,
• the number p of peaks, and
• the maximum value γ occurring in A.
Our main results are parameterized algorithms for the combined parameter (p, δ) in case of VPE and for the parameter δ in case of VE. For
the parameter maximum value γ, we show that VPE does not admit a
polynomial kernel unless NP ⊆ coNP/poly.
Next, by providing an integer linear programming formulation where the
number of variables is bounded in the number of peaks p and the maximum
difference δ, we prove fixed-parameter tractability with respect to them.
164
Theorem 6.3.1. Vector Positive Explanation parameterized by the
combined parameter number p of peaks and maximum difference δ is fixedparameter tractable.
Proof. We provide an integer linear program (ILP) formulation for VPE
where the number of variables is a function of p and δ. This ILP determines
whether there is a regular size-k explanation (this is correct by Theorem 6.2.3).
In a regular explanation the multiset of weights of segments that start at
an uptick sum up to the uptick size. Analogously, this holds for segments
ending at a downtick. Motivated by this fact, we introduce the following
notion: For a positive integer x, we say that
Pr a multiset X = {x1 , x2 , . . . , xr }
of positive integers partitions x if x = i=1 xi . Similarly, we say that X
partitions an uptick (downtick) i of size x if X partitions x. Let P(x) denote
the set of all multisets that partition x.
In the ILP formulation for VPE, we describe a solution by “fixing” for
each position i a multiset Xi of positive integers which partitions the uptick
(downtick) at i. The crucial observation for our ILP is that if a set of
consecutive upticks contains more than one uptick of size x, it is sufficient to
fix how many of these upticks were partitioned in which way. In other words,
one does not need to know the partition for each position; instead one can
distribute freely the partitions of x onto the upticks of size x. This also holds
for consecutive downticks. Since each peak is preceded by consecutive upticks
and succeeded by consecutive downticks, and since we introduce variables in
the ILP formulation to “model” how many upticks (downticks) exist between
two consecutive peaks, the number of variables in the formulation is bounded
by a function of p and δ. We now give the details of the formulation. Herein,
we assume that the peaks are ordered from left to right; we refer to the i-th
peak in this order as peak i.
For an integer x ∈ {1, . . . , δ}, let occ(x, i) denote the number of upticks of
size x that directly precede peak i, that is, the number of upticks succeeding
peak i − 1 and preceding peak i. Similarly, let occ(−x, i) denote the number
of downticks of size x that directly succeed i. For two positive integers
y and x with y ≤ x and a multiset P ∈ P(x) let mult(y, P ) denote how
often y appears in P . We use mult(y, P ) to “model” how many segments of
weight y start (end) at some uptick (downtick) that is partitioned by P .
To formulate the ILP, we introduce for each peak i, each x ∈ {1, . . . , δ},
and each P ∈ P(x) two nonnegative variables varx,P,i and var−x,P,i . The
variables respectively correspond to the number of upticks directly preceding
peak i and downticks directly succeeding peak i of size x that are partitioned
by P in a possible explanation of A. To enforce that a particular assignment
to these variables corresponds to a valid explanation, we introduce the
following constraints.
165
First, for each peak i and each 1 ≤ x ≤ δ we ensure that the number
of directly preceding size-x upticks (succeeding size-x downticks) that are
partitioned by some P ∈ P(x) is equal to the number of directly preceding
size-x upticks (succeeding size-x downticks):
X
∀i ∈ {1, . . . , p}, ∀x ∈ {−δ, . . . , δ} \ {0} :
varx,P,i = occ(x, i). (6.1)
P ∈P(x)
Second, we ensure that for each peak i and each value y ∈ {1, . . . , δ} the
number of segments of weight y that end directly after peak i is at most
the number of segments of weight y that start at positions (not necessarily
directly) preceding peak i minus the number of segments of weight y that end
at positions succeeding some peak j < i. Informally, this means that we only
“use” the available number of segments of weight y. To enforce this property,
for each peak 1 ≤ i ≤ p and each possible segment weight 1 ≤ y ≤ δ we add:
i X
δ
X
X
(
j=1 x=y P ∈P(x)
mult(y, P ) · varx,P,j
|
{z
}
# of started weight-y segments
− mult(y, P ) · var−x,P,j ) ≥ 0
|
{z
}
# of closed weight-y segments
(6.2)
Finally, we ensure that the total number of segments is at most k:
p X
δ
x
X
X X
mult(y, P ) · varx,P,i ≤ k.
(6.3)
i=1 x=1 P ∈P(x) y=1
Correctness: The equivalence of the ILP instance and (A, k) can be seen as
follows. Assume that there is a size-at-most-k explanation (I, ω) for (A, k),
where the segments start at upticks and end at downticks. Recall that by
definition of P(x), for any uptick i of size x there is a partition in P(x) that
corresponds to the weights of the segments starting in i. For each peak i, for
any value 1 ≤ x ≤ δ and each P ∈ P(x), count how many upticks of size x
that directly precede peak i are explained by segments in I (segments that
start in this uptick) whose weights correspond to P(x) and set varx,P,i to this
value. Symmetrically, do the same for the downticks succeeding peak i and
set var−x,P,i accordingly. It is straightforward to verify that Equations (6.1)
to (6.3) hold.
Conversely, assume that there is an assignment to the variables such that
Equations (6.1) to (6.3) are fulfilled. We form an explanation (I, ω) as follows:
For any peak i and any value 1 ≤ x ≤ δ with occ(x, i) > 0 let Pi,x be the
multiset of elements from P(x) that contains each P ∈ P(x) exactly varx,P,i
times. By Equation (6.1), |Pi,x | = occ(x, i). For an arbitrary ordering of Pi,x
166
and the upticks of size x directly preceding peak i, add to I for the jth
element Pj of Pi,x exactly |Pj | segments with weight corresponding to Pj
and let them start at the jth uptick with size x that directly precedes peak i.
By Equation (6.3) we added at most k segments. It remains to specify the
end of the segments. Symmetrically to the upticks, for each downtick directly
succeeding peak i of size x let Pi,x be the multiset of elements from P(x)
containing each P ∈ P(x) exactly var−x,P,i times. For the jth element Pj
of Pi,x and the jth downtick directly succeeding peak i (again both with
respect to any ordering) and for each α ∈ Pj pick any weight-α segment
from I (so far without end) and let it at the jth downtick. Observe that the
existence of this segment is ensured by Equation (6.2). Finally, it remains to
argue that the end of each segment in I is determined. This follows from
the fact that Equations (6.1) & (6.2) together imply for each 1 ≤ y ≤ δ that
p X
δ
X
X
(mult(y, P ) · varx,P,i − mult(y, P ) · var−x,P,i ) = 0,
i=1 x=y P ∈P(x)
and thus the total number of opened weight-y segments is equal to the
number of closed weight-y segments.
Running time: The ILP can be solved within the following time bound.
The number of variables in the constructed ILP instance is
p·
X
x∈{−δ,...,δ}\{0}
|P(|x|)| = 2p
δ
X
|P(x)| ≤
x=1
2δp · |P(δ)| ≤ 2δp · eπ
√2
3δ
= f (δ, p),
where the last inequality is due to Azevedo Pribitkin [Aze09].
Then, due to a deep result in combinatorial optimization, there is a
O(f (δ, p)2.5f (δ,p)+o(f (δ,p)) · |L|)-time algorithm deciding the feasibility of the
ILP, where |L| is the size of the instance [FT87, Kan87].
√ 2 Moreover, as we
have O(δp) inequalities, we also have |L| = O(δ 2 p2 · eπ 3 δ ).
Observe that Theorem 6.3.1 implies that VE is fixed-parameter tractable
with respect to the max. difference δ: By Corollary 6.2.5(ii) and Corollary 6.2.7(i), in linear time one can transform input instances of VE into
equivalent single-peaked instances of VPE without increasing the maximum
difference δ.
Corollary 6.3.2. Vector Explanation parameterized by the maximum
difference δ is fixed-parameter tractable.
167
It remains open whether VPE is fixed-parameter tractable with respect to δ.
Note that the argumentation for VE (Corollary 6.3.2) cannot be transferred,
since there may be more than one peak in a given instance. However, the
following theorem implies that there is a XP-algorithm (Definition 2.3.3)
for VPE parameterized by maximum difference δ. It is based on dynamic
programming.
Theorem
6.3.3. Vector Positive Explanation is solvable in O(nδ ·
√
π 2δ/3
) time.
e
Proof. We describe a dynamic programming algorithm that finds a regular
minimum-size explanation. Every explanation for a size-n vector A can be
interpreted as an extension of an explanation for the same vector without the
last entry, where some segments that originally only covered position n − 1
may be stretched to also cover position n and some new length-one segment
may start at position n.
Our algorithm uses the above relation between explanations for the vector A[1, . . . , n] and explanations for the vector A[1, . . . , n − 1]. Due to
Theorem 6.2.3, it only considers regular explanations, implying that each
segment starts at an uptick and ends at a downtick. Since all upticks and
downticks have size at most δ, the algorithm furthermore only considers
solutions in which all segments have weight at most δ.
We fill a table T which has entries of type T (i, d1 , . . . , dj , . . . , dδ ) where
0 ≤ i ≤ n and 0 ≤ dj ≤ k with 1 ≤ j ≤ δ. An entry T (i, d1 , . . . , dj , . . . , dδ )
contains the minimum number of segments explaining vector A[1, . . . , i] such
that dj segments of weight j cover position i. If no such explanation exists,
then the entry is set to ∞. By definition of the table entries, there is a
solution for VPE if and only if
min
(d1 ,...,dδ )∈{0,...,k}δ
T (n, d1 , . . . , dδ ) ≤ k.
In the following, we show how to fill the table. As initialization, set
T (0, d1 , . . . , dδ ) ← ∞ if there is some dj > 0 and set T (0, 0, . . . , 0) ← 0.
For increasing i ≤ n, compute the table for each (d1 , . . . , dδ ) ∈ {0, . . . , k}δ
Pδ
as follows. If A[i] = j=1 dj · j and A[i] > A[i − 1], then set

T (i, d1 , . . . , dδ ) ←
min
d01 ≤d1 ,...,d0δ ≤dδ

δ
X
T (i − 1, d01 , . . . , d0δ ) +
(dj − d0j ) .
j=1
(6.4)
168
If A[i] =
Pδ
j=1
dj · j and A[i] < A[i − 1], then set
T (i, d1 , . . . , dδ ) ←
min
d01 ≥d1 ,...,d0δ ≥dδ
T (i − 1, d01 , . . . , d0δ ).
(6.5)
Otherwise, set
T (i, d1 , . . . , dδ ) ← ∞.
(6.6)
The correctness of the initialization follows directly from the table definition. For the remaining computation we can thus assume that there is
some i such that all entries T (i0 , d1 , . . . , dδ ) with (d1 , . . . , dδ ) ∈ {0, . . . , k}δ
and i0 < i were computed correctly.
As discussed above, we interpret an explanation of A[1, . . . , i] as extension
of an explanation for A[1, . . . , i−1]. There are exactly two groups of segments
covering position i: those also covering position i − 1 and those starting
at position i. Let the set of segments
covering position i be described
Pδ
by (d1 , . . . , dδ ) such that A[i] = j=1 dj · j and A[i] > A[i − 1]. Due to
Theorem 6.2.3, no segment ends at position i, but since A[i] > A[i − 1] at
least one new segment has to start at position i. By setting (d01 , . . . , d0δ )
such that d0j ≤ dj , 1 ≤ j ≤ δ, one considers all possible extensions for
explanations of A[i − 1] such that no segment ends at position i. Clearly,
Pδ
0
j=1 (dj − dj ) further segments have to start at position i to explain A[i].
Hence, assignment (6.4) is correct.
Now, describe the set of segments covering position i by (d1 , . . . , dδ ) such
Pδ
that A[i] = j=1 dj ·j and A[i] < A[i−1]. By Theorem 6.2.3 no new segment
starts at position i. The algorithm considers all possible explanations where
some segments end at position i and the other segments survive to explain A[i].
Thus, assignment (6.5) is correct.
For a given (d1 , . . . , dδ ) ∈ {0, . . . , k}δ , to find an explanation for A[1, . . . , i]
Pδ
such that A[i] 6= j=1 dj · j is impossible because such an explanation does
not explain position i. Thus assignment (6.6) is correct.
The size of the table is upper-boundedPby nδ since we only have to consider
δ
table entries T [i, d1 , . . . , dδ ] with A[i] = j=1 (dj ·j). The trivial upper bound
of O(nδ ) for computing each table entry already leads to a running time
of O(n2δ ). However, the number of entries that have to be considered
is smaller. For assignment (6.4), one only has to consider those entries of
Pδ
Table T that do not have value ∞. Hence, j=1 |dj −d0j | ≤ |A[i]−A[i−1]| ≤
δ. This implies that for each table entry the number of previous entries
that have to be considered in the minimization is upper-bounded by the
number √
of different multisets that sum up to δ and thus is upper-bounded
2
by O(eπ 3 δ ) [Aze09]. A similar argument applies for assignment (6.5). The
overall running time follows.
169
It is known that VPE is fixed-parameter tractable with respect to the
parameter maximum value γ [Bie+13]. We complement this result by showing
a lower bound on the kernel size, and thus demonstrate limitations on the
power of polynomial-time preprocessing.
Theorem 6.3.4. Unless NP ⊆ coNP/poly, there is no polynomial kernel for
Vector Positive Explanation parameterized by the maximum value γ.
Proof. We provide an AND-cross-composition (Definition 2.3.10) from the
3-Partition problem.
3-Partition [GJ79](SP15)
Input: A multiset S = {a1 , . . . , a3m } of positive integers and an integer
P3m
bound B with m · B = i=1 ai and B/4 < ai < B/2 for every
i ∈ {1, . . . , 3m}.
Question: Is there
P a partition of S into m subsets P1 , . . . , Pm with |Pj | = 3
and ai ∈Pj ai = B for every j ∈ {1, . . . , m}?
3-Partition is NP-complete even if B (and thus all ai ’s) is bounded by a
polynomial in m [GJ79](SP15). We show that this variant of 3-Partition
AND-cross-composes to VPE parameterized by the maximum value γ. Then,
Theorem 2.3.11 imply that VPE does not have a polynomial kernel with
respect to parameter γ, unless NP ⊆ coNP/poly.
First, let (S, B) be a single instance of 3-Partition. We show that it
reduces to an instance (A0 , 3m) of VPE. This reduction is very similar to a
previous NP-completeness reduction for VPE due to Bansal et al. [Ban+11].
We define A0 as length-(4m − 1) vector:
!
j
3m
X
X
a1 , a1 + a2 , . . . ,
ai , . . . ,
ai = mB, (m − 1)B, (m − 2)B, . . . , B .
i=1
i=1
On the one hand, if a partition P1 , . . . , Pm of S forms a solution, then the
set of segments {[i, 3m + j] | ai ∈ Pj } each with weight w([i, 3m + j]) = ai is
an explanation for the vector A0 . On the other hand, let (I, w) be a regular
explanation for (A0 , 3m). Since every segment starts at an uptick and ends
at a downtick, I contains 3m segments and the segment starting at position i
has weight ai . Since B/4 < ai < B/2 for each integer ai ∈ S, exactly three
segments end at a downtick whose size is exactly B. Thus, grouping the
segments according to the position they end at, we get the desired partition
of S, solving the instance of 3-Partition.
Now let (S1 , B1 ), . . . , (St , Bt ) be instances of the 3-Partition problem
such that Sr = {ar1 , . . . , ar3mr } and Br ≤ mr c for every r ∈ {1, . . . , t} and
some constant c. We build an instance (A, k) of VPE by first using the
170
above reduction for each (Sr , Br ) separately to produce a vector A0r , and
then concatenating the vectors A0r one after another, leaving a single
Pt position
of value 0 in between. The total length of the vector A is 4( r=1 mr ) − 1
Pt
and we set k = 3 r=1 mr .
Due to the argumentation for the single instance case, on the one hand, if
each of the instances is a yes-instance, thenP
there is an explanation using
t
3mr segments per instance (Sr , Br ), that is 3 r=1 mr segments in total. On
the other hand, we need at least 3mr segments to explain A0r and there is an
explanation with 3mr segments if and only if (Sr , Br ) is a yes-instance. Since
all segments are positive and the subvectors A0r are separated by a position
with value zero, no segment can span over two subvectors. In other words,
no segment can be used to explain
more than one of the A0r ’s. Therefore,
Pt
an explanation for A with 3 r=1 mr segments implies that (Sr , Br ) is a
yes-instance for every r ∈ {1, . . . , t}.
Finally, observe that the maximum value γ in the vector A is equal
to maxtr=1 mr Br ≤ maxtr=1 mr c+1 and, thus, it is polynomially bounded
in maxtr=1 |Sr |. Hence, 3-Partition AND-cross-composes to VPE parameterized by the maximum value γ, and thus by Theorem 2.3.11 there is no
polynomial kernel for VPE unless NP ⊆ coNP/poly.
6.4 Parameterizations of the Size and the
Structure of Solutions
We now provide fixed-parameter tractability and (parameterized) hardness
results for further natural parameters. Specifically, we consider the number k
of segments in the solution, so-called “above-guarantee” and “below-guarantee”
parameterizations (which are smaller than k), the maximum segment length ξ,
and the maximum number of segments covering a position.
For the parameter k we obtain fixed-parameter tractability by using
Reduction Rule 6.1.1 (p. 156), Corollary 6.2.5, and Corollary 6.2.7 (both
Section 6.2) to develop search tree algorithms for VPE and VE. The depth
and the branching degree of the search tree are bounded by the solution
size k. Recall that after an exhaustive application of Reduction Rule 6.1.1
every input vector has at most 2k − 1 entries, implying a (2k − 1)-entry
kernel (Lemma 6.1.2).
Theorem 6.4.1. Vector Positive Explanation and Vector Explanation can be solved in O(k! · k + n) time.
Proof. We start with the algorithm for VPE which works as follows. After
exhaustive application of Reduction Rule 6.1.1 branch over all possible
171
segments covering the last entry. Due to Corollary 6.2.5(i), it suffices to
search for exactly one segment starting at one of the upticks and ending
at the last entry. For each branch assign the value A[n] as weight to the
segment and solve the instance consisting of the remaining entries recursively.
To this end, decrease each of the entries covered by the segment by A[n], and
decrease k by one. Whenever an entry becomes negative discard the branch.
Clearly, the search tree produced by the branching algorithm has depth at
most k. In the i-th level of the search tree, one branches over at most k +1−i
upticks (Corollary 6.2.5). After exhaustively applying Reduction Rule 6.1.1
once in O(n) time, the steps performed in each search tree node take O(k)
time. The overall running time thus is O(k! · k + n).
For VE we first apply Corollary 6.2.7(i) to transform our instance into a
single-peaked instance (this is necessary to avoid negative entries). The rest
works analogously to VPE.
The second part of Theorem 6.4.1 implies that for a reduced instance
every explanation needs at least bn/2c + 1 segments. Furthermore, instances
with k = bn/2c + 1 are solvable in polynomial time (below, we will state a
generalization of this fact). Hence, it is interesting to study parameters that
measure how far we have to exceed this lower bound for the solution size;
notably, such above-guarantee parameters can be significantly smaller than k.
For this reason, we study a parameter that measures k − (bn/2c + 1). For
ease of presentation, we define this parameter as φ = 2k − n. The concepts
of “clean” and “messy” positions, which are defined as follows, are crucial for
the design of our algorithms.
Definition 6.4.2 (Clean and Messy Segments). Let (A, k) be an instance of
Vector Explanation or Vector Positive Explanation and let I be
an explanation for A. A segment I = [i, j] ∈ I is clean if all other segments
start/end at positions different from i/j. A position i is clean with respect
to I if it is the start or endpoint of a clean segment in I. A position or
segment that is not clean is called messy.
Remark. Note that corresponding to messy overlapping configurations (Figure 6.3) a segment is messy if and only if it is in a messy overlapping
configuration with any other segment.
We show (Theorem 6.4.6(i)) that clean positions can always be covered
by clean segments of “minimum length”: For an input vector A with the
corresponding tick vector T , iterate from left to right over all clean positions
and for each position i (still clean) find the first clean position j > i with
T [i] = −T [j] and add a weight-T [i] segment [i, j].
Additionally, we show that for every yes-instance of VE, the number of
messy positions is at most 3φ and the number of messy segments used by
172
j
i
i
I
h
H
I
`
h
H
0
`
j
i
0
I
i
I
j
h
H
`
0
`
h
H
j
0
Figure 6.4: Illustration of the proof of Lemma 6.4.3. The two equal-weight segments
H = [h, l] and I = [i, j] with i < l < j are replaced by I 0 = [i, l] and H 0 = [h, j]
without changing the weight. The two different cases whether h > i (left) or not
(right) are illustrated.
an explanation is at most 2φ with φ = 2k − n. Furthermore, if there are
an uptick and a downtick of the same size in a single-peaked instance, then
we may assume that the corresponding segment is contained in the solution
(Lemma 6.4.5).
As the following theorem shows, using the above mentioned properties
concerning clean and messy positions, we can replace the exponent k in
the running time of Theorem 6.4.1 by the smaller φ. This also implies that
VPE is polynomial-time solvable for constant φ. Unless W[1] = FPT, this
result cannot be improved to fixed-parameter tractability since we give a
parameterized reduction from the W[1]-hard Subset Sum problem [FK93]
to VPE. In contrast, VPE for single-peaked instances as well as VE in
general are fixed-parameter tractable with respect to φ and can be efficiently
reduced to equivalent instances with at most 3φ positions.
Towards proving the above mentioned results, we first show some combinatorial properties of clean and messy positions (Definition 6.4.2).
Lemma 6.4.3. Let I be an explanation for an instance (A, k) of Vector
Positive Explanation. Furthermore, let [i, j] ∈ I be a clean segment and
let ` be a clean position such that i < ` < j and ` and j are downticks of
the same size. There is an explanation I 0 for (A, k) containing the clean
segment [i, `] such that |I 0 | = |I|.
Proof. Since ` is clean, there is a clean segment H = [h, `] whose weight
equals the downtick size of ` and hence the weight of I = [i, j]. Consider
a segment set I 0 obtained from I by replacing I with I 0 = [i, `] and H
with H 0 = [h, j]. The weights remain unchanged. In particular, the weights
of I 0 and H 0 are the same as the weights of I and H. It is straightforward to
verify by a case distinction whether h > i or not, that I 0 with the adjusted
weight function is still an explanation. See Figure 6.4 for an illustration of
the two cases.
173
Lemma 6.4.4. Let (A, k) be a yes-instance of Vector Positive Explanation that is reduced with respect to Reduction Rule 6.1.1. Then,
every explanation of (A, k) has at most 2φ messy segments and at most 3φ
messy positions.
Proof. Let x denote the number of messy segments in some arbitrary explanation for (A, k). Since (A, k) is reduced with respect to Reduction
Rule 6.1.1, every position of A is the starting point or endpoint of some
segment. In particular, every messy segment shares at least one endpoint
with another messy segment. Hence, there are at most 1.5x messy positions
in the explanation. Furthermore, there are at most 2(k − x) clean positions.
Thus, n ≤ 2(k − x) + 1.5x which implies x ≤ 2φ, and the number of messy
positions is at most 1.5x ≤ 3φ.
Lemma 6.4.5. Let (A, k) be a single-peaked instance of Vector Positive
Explanation. If vector A has an uptick i and a downtick j of the same
sizes, then there is an explanation for (A, k) containing the segment [i, j]
with weight equal to the size of the uptick i.
Proof. Let (A, k) be a single-peaked VPE instance and let T be the tick
vector of A. By Corollary 6.2.5(ii) it is an equivalent VE instance and thus
by Theorem 6.2.6 we may assume that j = i + 1. Furthermore, let (I, w) be
a regular explanation of (A, k). Let Is be all segments in I starting in i and
let Ie be all segments in I ending in i + 1 and, additionally, do not start in i.
Hence Is ∩ Ie = ∅.
Let T 0 be a copy of T and “subtract” the segments in I \ (Is ∪ Ie ): For each
segment [`, r] ∈ I \ (Is ∪ Ie ) of weight a decrease T 0 [`] by a and increase T 0 [r]
by a. Additionally, subtract [i, i + 1] with weight T [i], meaning that we
set T 0 [i] and T 0 [i + 1] from ±T [i] to zero. Clearly, there is an explanation
for (A, k) containing [i, i + 1] of weight T [i] if there is a size |Is | + |Ie | − 1
explanation for the input vector corresponding to T 0 . This holds since we
subtracted all segments in I \ (Is ∪ Ie ) and thus in T 0 only positions equal
to the start or end of segments in Is ∪ Ie may be different from zero. Thus,
since we additionally inserted [i, i + 1], there are at most |Ie | upticks and at
most |Is | downticks in T 0 . Hence, the corresponding input vector of T 0 has,
after an application of Reduction Rule 6.1.1, at most |Ie |+|Is |−1 entries.
We now have all ingredients to provide our (in-)tractability results with
respect to the parameter φ.
Theorem 6.4.6. (i) Vector Positive Explanation can be solved in
O((2k)3φ · (2φ)! · φ · k + k log k + n) time.
174
(ii) Any single-peaked instance of Vector Positive Explanation and
any instance of Vector Explanation can be reduced in O(n +
k log k) time to an equivalent one with most 3φ entries. Moreover,
Vector Explanation and single-peaked Vector Positive Explanation are solvable in O((2φ)! · φ + k log k + n) time.
Proof. Theorem 6.4.6(i): We prove that VPE can be solved in O((2k)3φ ·
(2φ)! · φ · k + k log k + n) time. Let (A, k) be an instance of VPE and let T
be the tick vector corresponding to A. We may assume via a preprocessing
step running in O(n) time that Reduction Rule 6.1.1 has been exhaustively
applied and thus the number of positions is at most 2k.
The algorithm works as follows. Let U (D) be the set of all upticks
(downticks) in T . Sort the values in U and D in ascending order according
to their absolute size and use their position in T as a tie-breaker (smaller
positions come first). This can be done in O(k log k) time. Next, branch
into the at most (2k)3φ possibilities for choosing all of the at most 3φ messy
positions (Lemma 6.4.4). If the guess was correct, then for each clean uptick
there is clean downtick of equal size.
By Lemma 6.4.3 there is a minimum-size explanation that contains a
segment starting in any clean uptick position i and ending at the first clean
downtick position j > i with the same size. We next find and remove these
segments: Initialize k̃ by the value of k and also T 0 by T . Iterate over all
clean upticks in the order of U and find for each of them the first clean
downtick in D which starts to its right. Delete the up- and downtick from T 0
and decrease parameter k̃ by one. Clearly, by using two pointers, one for U
and one for D, iterating over U and finding the downtick in D can be done
in O(k) time as by the order of U and D one has to move the pointers only
to the right. Moreover, if at some point of the iteration we do not find
any “matching” downtick or at the end there remain some clean downticks
in D, then we abort this branch as the guess of clean positions was incorrect.
Let A0 be the input vector corresponding to the final T 0 . Note that since
all positions in A0 are messy, by Lemma 6.4.4 it follows that |A0 | ≤ 2φ
and k̃ ≤ 3φ. Hence, Theorem 6.4.1 solves the remaining instance (A0 , k̃) in
O((2φ)!·φ) time (A0 is already reduced with respect to Reduction Rule 6.1.1).
The overall running time is O((2k)3φ · (2φ)! · φ · k + k log k + n).
Theorem 6.4.6(ii): Due to Corollary 6.2.7(i), we can assume that the given
instance of VE is single-peaked. Also, because of Corollary 6.2.5(ii), we only
need to investigate whether the given single-peaked instance is a yes-instance
for VPE. We first apply Reduction Rule 6.1.1 exhaustively. After that, if
there is an uptick and a downtick of the same size, then by Lemma 6.4.5
there is an optimal solution containing a segment starting at the uptick and
ending at the downtick of weight equal to the size of the uptick. Hence,
175
by applying a similar procedure as in the proof of Theorem 6.4.6(i) (sort
up- and downticks by their size) one finds and eliminates all these segments
in O(k log k) time. Note that by removing such a segment from the input
vector the length of the vector is reduced by two, while k is reduced by one,
so φ stays the same.
In the remaining instance all positions are messy and thus by Lemma 6.4.4
there are at most 3φ messy positions and 2φ messy segments explaining
them. Thus, one ends up with a problem kernel having at most 3φ positions.
By Theorem 6.4.1, this kernel can be solved in O((2φ)! · 2φ + 3φ) time.
Theorem 6.4.6(ii) states that one single-peaked instances VPE is fixedparameter tractable with respect to φ. In difference to VE, we next show
that in general VPE is W[1]-hard when parameterized by φ.
Theorem 6.4.7. Vector Positive Explanation is W[1]-hard with respect to φ.
Proof. We present a parameterized reduction from the NP-complete Subset
Sum problem.
Subset Sum [GJ79](SP13)
Input: A multiset X = {x1 , . . . , x` } of positive integers and two positive integers y and Φ.
P
Question: Is there a size-Φ subset X 0 of X such that xi ∈X 0 xi = y?
Subset Sum is W[1]-hard
P with respect to the solution size Φ [FK93]. In
the following, we use t = 1≤i≤` xi to denote the total sum of the integers
in X. Note that by modifying
P the xi ’s we can assume that for every size(Φ − 1) subset X 0 the sum xi ∈X 0 xi is less than y: adding t to each input
integer, and Φ · t to y results in an instance for which this holds. Next, we
describe the parameterized reduction.
Pi
The input vector A has length 2` + 1. For i ≤ `, we set A[i] = j=1 xj .
Let A[` + 1] = t − y. For i ≥ ` + 1, we set A[i] = A[2` + 2 − i]. The number of
allowed segments is set to `+Φ. Consequently, φ = 2(`+Φ)−(2`+1) = 2Φ−1.
We complete the proof by showing that for this construction the following
equivalence holds.
(X, y, Φ) is a yes-instance of Subset Sum ⇔ (A, ` + Φ) is a
yes-instance of VPE.
“⇒”: Let X 0 be a size-Φ subset of X whose values sum up to y. Then,
consider the following set I of segments.
For each xi ∈
/ X 0 , add the segment Ji = [i, 2` + 3 − i]. There are
` − Φ such segments. For each xi ∈ X 0 , add two segments Ii = [i, ` + 1]
176
and Ii0 = [` + 2, 2` + 3 − i]. For each of these two types of segments there
are Φ of them. Hence, |I| = ` + Φ. For each 1 ≤ i ≤ ` set the weights of
the segments
Ji , Ii and Ii0 to xi . Now, I explains A: First, for each i ≤ `,
P
A[i] = j≤i xj is explained by {Jj | j ≤ i ∧ xj ∈
/ X 0 } ∪ {Ij | j ≤ i ∧ xj ∈ X 0 }.
Second, A[` + 1] = t − y is explained by exactly P
the segments Jj with xj ∈
/ X 0.
Finally, for i > ` + 1, A[i] = A[2` + 2 − i] = j≤2`+2−i xj is explained by
{Jj | j ≤ 2` + 2 − i ∧ xj ∈
/ X 0 } ∪ {Ij0 | j ≤ 2` + 2 − i ∧ xj ∈ X 0 }.
“⇐”: Let I be a set of ` + Φ segments that explain A. By Theorem 6.2.3
we can assume that I is regular. First, note that for each position i ≤ `,
there is at least one segment that starts at i. Also, each of these segments
0
0
has a weight of
Pat most the maximum in X. Since for any X with |X | < Φ
it holds that xi ∈X 0 xi < y and the size of downtick ` + 1 is y, at least Φ
segments end at ` + 1. Similarly, for each i ≥ ` + 3 there is at least one
segment that ends at position i. Each of these segments has a weight of
at most xj for some xj ∈ X. Further, since the size of uptick ` + 2 is y,
at least Φ segments start at ` + 2. This implies that there are exactly `
segments starting in the first ` positions and exactly Φ segments ending at
position ` + 1. Therefore, for each i ≤ ` there is exactly one segment starting
at i which has weight xi . Since Φ of these segments end at position `+1, they
0
correspond
P to a size-Φ set X ⊆ X. Finally, since A[`] = t and A[`+1] = t−y
the sum xi ∈X 0 xi of the integers in this set is exactly y.
Parameter φ used in Theorems 6.4.6 & 6.4.7 measures how far the solution
exceeds the lower bound bn/2c + 1. Another bound on the solution size
is n: If k = n, then any instance of VPE or VE is a trivial yes-instance.
Hence, it is interesting to consider the parameter n − k. Furthermore, it
is natural to consider explanations with restricted segment length ξ or the
maximum number o of segments overlapping at some position. The following
theorem shows that VPE and VE are already NP-complete even if k = n − 1,
ξ ≥ 3, and o = 2. To this end, we reduce from the NP-complete Partition
problem [GJ79](SP12). In terms of parameterized complexity this implies
that, unless P = NP, VPE is not fixed-parameter tractable with respect
to the “maximum segment length ξ”, the “maximum number o of segments
overlapping at some position”, and the “below guarantee parameter” n − k.
Theorem 6.4.8. Vector Positive Explanation and Vector Explanation are NP-complete even if k = n − 1 and every yes-instance has an
explanation of at most k segments where each position is covered by at most
two segments and each segment has length at most three.
Proof. We reduce from the NP-complete Partition problem.
177
Partition [GJ79](SP12)
Input: A multiset of positive integers S = P
{a1 , . . . , at }.P
Question: Is there a subset S 0 ⊆ S such that ai ∈S 0 ai = ai ∈S\S 0 ai ?
Given an instance S = {a1 , . . . , at } of Partition, we create an input
instance (A, k), where A is a vector of length 3t + 1 and k = 3t. Namely, AT
is the vector


1


2




2 + (t + 1)a1




3 + (t + 1)a1




4 + (t + 1)a1



4 + (t + 1)(a1 + a2 ) 


..




.


P
 2j − 1 + (t + 1) j−1 ai

i=1


P
j−1


2j + (t + 1) i=1 ai


Pj


2j + (t + 1) i=1 ai




..


.


Pt−1



 2t − 1 + (t + 1) i=1 ai


Pt−1


2t + (t + 1) i=1 ai


P
t


2t + (t + 1) i=1 ai
P
t
1
t + 2 (t + 1) · i=1 ai
Obviously, the reduction runs in polynomial time. It remains to show that
S = {a1 , . . . , at } is a yes-instance of Partition ⇔ (A, k = 3t) is
a yes-instance of VPE and VE.
0
“⇒”:
S be a solution for the Partition instance, meaning
P Let S ⊆ P
that ai ∈S 0 ai = ai ∈S\S 0 ai . Further, let Sj0 = S 0 ∩ {a1 , . . . , aj }, S j =
{a1 , . . . , aj } \ S 0 and S00 = S 0 = ∅. We construct the set I of segments
consisting of six subsets and their weights as follows (we use the notation
[l, r; a] for a weight-a segment starting at l and ending at r):
X
I1 = {[3j − 2, 3j + 1; j + (t + 1) ·
ai ] | aj ∈
/ S 0 },
0
ai ∈Sj−1
I2 = {[3j − 1, 3j; j + (t + 1) ·
X
ai ] | aj ∈
/ S 0 },
ai ∈S j−1
I3 = {[3j, 3j + 2; j + (t + 1) ·
X
ai ∈S j
178
ai ] | aj ∈
/ S 0 },
X
I4 = {[3j − 1, 3j + 2; j + (t + 1) ·
ai ] | aj ∈ S 0 }, ,
ai ∈S j−1
I5 = {[3j − 2, 3j; j + (t + 1) ·
X
ai ] | aj ∈ S 0 },
0
ai ∈Sj−1
I6 = {[3j, 3j + 1; j + (t + 1) ·
X
ai ] | aj ∈ S 0 }.
ai ∈Sj0
As there are exactly three segments for each aj , there are 3t segments in total.
0
Note that if aj ∈
/ S 0 , then Sj−1
= Sj0 . Otherwise aj ∈
/ S 0 and S j−1 = S j .
Now, we show that I with weight function w explains vector A. Let j ∈
{1, . . . , t}. At position 3j − 2 = 3(j − 1) + 1 we have A[3j − 2] = 2j − 1 +
Pj−1
(t + 1) i=1 ai . If aj ∈
/ S 0 , then segment [3j − 2, 3j + 1] from I1 covers
0
3j − 2 and if aj ∈ S , then segment
P [3j − 2, 3j] from I5 covers 3j − 2.0 Both
segments have weight j + (t + 1) ai ∈S 0 ai . Additionally, if aj−1 ∈
/ S , then
j−1
segment [3(j − 1), 3(j − 1) + 2] from I3 also covers 3j − 2 and if aj−1 ∈ S 0 ,
then segment [3(j − 1) − 1, 3(j − 1) + 2] from I4 P
also covers 3j − 2. In both
cases the weight of the segment is (j − 1) + (t + 1) ai ∈S j−1 ai . In the former
case this holds by definition. In the latter case, since aj−1 ∈ S 0 , it holds that
aj−1 ∈
/ S 0 and, thus, S 0 j−2 = S 0 j−1 . Summarizing, in each case the weights
of the two segments covering position 3j − 2 sum up to

j + (t + 1)

X


X
ai  + (j − 1) + (t + 1)
0
ai ∈Sj−1
ai  =
ai ∈S j−1
2j − 1 + (t + 1)
j−1
X
ai = A[3j − 2].
i=1
Pj−1
In the same way, at position 3j − 1, we have A[3j − 2] = 2j + (t + 1) i=1 ai .
If aj ∈
/ S 0 , then only segments [3j − 2, 3j + 1] from I1 and [3j − 1, 3j] from I2
cover and explain this position, since

j + (t + 1)

X
0
ai ∈Sj−1


ai  + j + (t + 1)
X
ai 
ai ∈S j−1
= 2j + (t + 1)
j−1
X
ai = A[3j − 1].
i=1
Otherwise, only segments [3j − 1, 3j + 2] from I4 and [3j − 2, 3j] from I5
179
cover and explain this position, since


j + (t + 1)
X


ai  + j + (t + 1)
X
ai 
0
ai ∈Sj−1
ai ∈S j−1
= 2j + (t + 1)
j−1
X
ai = A[3j − 1].
i=1
Pj
Also, at position 3j, we have A[3j] = 2j + i=1 ai . If aj ∈
/ S 0 , then only
segments [3j − 2, 3j + 1] from I1 and [3j, 3j + 2] from I3 cover and explain
this position since the sum of their weights equals

 

j
X
X
X
j + (t + 1)
ai  + j + (t + 1)
ai  = 2j + (t + 1)
ai = A[3j].
ai ∈Sj0
ai ∈S j
i=1
This also holds for the case that aj ∈ S 0 . Finally, we have only one segment
covering the position 3t + 1 with weight
t + (t + 1)
X
ai ∈S t
ai = t + (t + 1)
X
ai ∈S\S
t
X
1
ai = A[3t + 1].
ai = t + (t + 1) ·
2
0
i=1
“⇐”: Let I with weights w be a regular explanation for vector A with at
most k segments. As all upticks precede all downticks, all segments in I are
positive. More precisely, as there are exactly k = 3t upticks, exactly one
positive segment starts at every uptick and ends either at position 3t + 1 or
3t + 2.
We denote the segment of I starting at position 3i by Ii . Obviously,
w(Ii ) = (t + 1) · ai . Furthermore, there are 2t segments of weight one. Now
set S 0 = {ai | Ii ends at position 3t + 2}. We show that S 0 is a solution of
the Partition instance S: Let x ∈ {0, . . . , 2t} be theP
number of segments of
weight 1 that cover position 3t+1. We have x+(t+1) ai ∈S 0 ai = A[3t+1] =
Pt
P
Pt
t + 12 (t + 1) · i=1 ai . As |t − x| ≤ t, we have ai ∈S 0 ai = 12 i=1 ai . Hence,
0
S is a solution for the Partition instance S.
As we can see from the reduction, every yes-instance of Partition is
reduced to a yes-instance that can be explained by segments with ξ = 3 and
o = 2 and every no-instance is reduced to an instance that cannot be explained
by segments of any size. The statement of Theorem 6.4.8 follows.
We show that, in contrast to the NP-completeness for ξ ≥ 3, VPE and
VE are polynomial-time solvable for ξ ≤ 2.
180
Theorem 6.4.9. Vector Explanation and Vector Positive Explanation can be solved in O(n2 ) time for maximum segment length ξ = 2.
Proof. We devise a dynamic programming algorithm for VPE. Afterwards,
we show how to extend our algorithm to VE.
Let (A, k) be an input instance, where A is a vector of length n. Without
loss of generality, the last position is only covered by either one length-two
segment or one length-one segment, but not both, because then we can
always transform it into two length-one segments. Due to this, if we have
an optimal solution for a vector of length x, then we can find an optimal
solution for a vector of length x + 1 which contains either an additional
length-one segment or a length-two segment covering the last position.
Based on this idea, we use dynamic programming with a table s indexed by
1, . . . , n: For each j ≤ n, we store in s[j] the minimum number of segments
needed to explain the subvector A[1, . . . , j]. Let s[0] = 0 for simplicity. We
start with j = 1. For j = 1, we set s[1] = 1 which is obviously correct. Now
assume that for an index j ≤ n, s[i] was already computed for each i < j
and we now compute s[j]. We begin with i = j and aji = A[j]. We set
aji−1 = A[i − 1] − aji and i = i − 1
as long as
aji > 0 and i > 1.
(*)
The idea behind this computation is that if we want to add some lengthtwo segment with weight aji covering positions i and i − 1, then we should
make sure that after this the remaining value at position i − 1 is nonzero, since otherwise it is better to just cover position i separately. If
Condition (*) does not hold, then there are two cases: If aji = 0, then let
s[j] = min{s[j −1]+1, s[i−1]+j −i}; otherwise let s[j] = s[j −1]+1. Finally,
once the table is completed, we answer yes if s[n] ≤ k, and no otherwise.
As the algorithm obviously works in O(n2 ) time, it remains to show
that the algorithm fills the table correctly. The proof is by induction on j.
Obviously s[1] is computed correctly. For j ≤ n, assume s[i] is optimal for
all i < j. We show that s[j] is also optimal.
Let us first show that there is an explanation for A[1, . . . , j] with s[j]
segments. We have two cases: If s[j] = s[j − 1] + 1, then we use the
explanation for A[1, . . . , j − 1] with s[j − 1] segments and add a single lengthone segment to explain A[j]. Otherwise, there is an i ∈ {1, j − 1} such that
s[j] = s[i − 1] + j − i. Let ajj = A[j], and ajx = A[x] − ajx+1 for i ≤ x ≤ j − 1.
Note that aji = 0 because of Condition (*). Then, we use the explanation
for A[1, . . . , i − 1] with s[i − 1] segments and add a set I of j − i length-two
segments such that for each z ∈ {i, . . . j −1}, we have a segment Iz = [z, z +2]
181
with weight ajz+1 . Clearly, positions from 1 to i − 1 are already explained.
Since aji equals zero, we have A[i] = aji+1 which is also the weight of Ii .
Thus, I explains A[i]. For z ∈ {i + 1, . . . , j − 1}, we have A[z] = ajz + ajz+1
and A[j] = ajj . Hence, the subvector A[i + 1, . . . , j] is also explained by I.
Next, we show that s[j] is optimal. Suppose that there is an explanation
(I, w) of A[1, . . . , j] with r segments. We will show that r ≥ s[j]. Without loss
of generality, we can assume that every length-one segment exclusively covers
a position, since otherwise we can either merge two length-one segments or
split one length-two segment into two length-one segments and merge one of
them with the original length-one segment. We also assume that entry A[j]
is positive as otherwise s[j] = s[j − 1] ≤ r. Let i be the last position such
that all segments in I covering i start at i. If i = j, then I \ {[j, j + 1]} is an
explanation for A[1, . . . , j −1], and r ≥ s[j −1]+1 ≥ s[j] as s[j −1] is optimal.
If i < j, then I contains a chain of j − i overlapping length-two segments
Ii+1 = [i, i + 2], . . . , Ij = [j − 1, j + 1] starting at i and ending at j + 1. Since
these are the only segments explaining positions i, . . . , j, their weights are
w(Ij ) = A[j] and w(Iz ) = A[z] − w(Iz+1 ), j − 1 ≥ z ≥ i + 1. Position i is only
explained by Ii+1 , so we have A[i] = w(Ii+1 ) = A[i + 1] − w(Ii+2 ). Note that
applying the dynamic programming algorithm ajz = w(Iz ), i+1 ≤ z ≤ j. This
means that the algorithm stops at position i with aji = w(Ii+1 ) − aji+1 = 0.
Thus, s[j] = min{s[j − 1] + 1, s[i − 1] + j − i} ≤ s[i − 1] + j − i. Furthermore,
I \ {Iz | z ∈ {i + 1, . . . j}} is an explanation for A[1, . . . , i − 1]. Hence,
r ≥ s[i − 1] + j − i ≥ s[j] because s[i − 1] is optimal.
To solve VE, it is enough to change Condition (*) in the loop of the above
algorithm to “. . . as long as aji 6= 0 and i > 1”. The rest of the proof remains
the same.
6.5 Conclusion and Open Questions
We explored the parameterized complexity of Vector (Positive) Explanation with respect to various parameterizations. By considering the
tick vector concept, we gained further combinatorial insights into Vector
(Positive) Explanation. In particular, we could show that for Vector
Explanation the tick vector can be arbitrarily reordered. Several of our
parameterized algorithms for Vector (Positive) Explanation are based
on this observation. Furthermore, we found that, surprisingly, Vector Positive Explanation is harder than Vector Explanation, for example
with respect to the distance from triviality parameter φ = 2k − n.
It would be interesting to significantly improve on several of the running
time upper bounds of our (theoretical) tractability results (cf. Table 6.1 for
182
an overview). In particular, obtaining tight lower and upper running time
bounds for the parameter number k of segments seems to be a challenging
and interesting research task. Moreover, we also left open a number of
concrete problems:
• Is Vector Positive Explanation fixed-parameter tractable with
respect to the maximum difference δ?
• Does Vector Explanation parameterized by δ or parameterized
by γ admit a polynomial kernel?
• Is Vector Explanation or Vector Positive Explanation fixedparameter tractable with respect to the parameter “number of different
values in the input vector A”? This parameter would be a natural
version of “parameterization by the number of numbers” [FGR12].
Last but not least, we would like to point to the challenging task to transfer
our study to the case of a 2-dimensional (“matrix”) input [Kar+11].
183
7 Conclusion and Outlook
In this thesis we studied three approaches to systematically identify tractable
cases of NP-hard problems. We have considered the approach of restricting
a graph problem to special graph classes, to consider structural parameterizations, and the approach to consider special neighborhood structures in
local search. We presented for each of these three approaches a case study
on a graph problem and further applied the second approach of structural
parameterizations to a non-graph problem. In this section we briefly recapitulate our main findings presented in Chapters 3 to 6 and we outline future
research directions which would canonically extend our work.
We applied the first approach of considering restricted graph classes to the
Metric Dimension problem (Chapter 3). We have shown that even on the
highly restricted graph class of bipartite graphs with maximum degree three
Metric Dimension is W[2]-hard with respect to the solution size parameter.
The corresponding reduction also yields the inapproximability result of
excluding the existence of a polynomial-time factor-o(log n) approximation
for Metric Dimension, unless P = NP.
Following the same approach, a natural follow up challenge, which is
motivated by the NP-hardness proof for planar graphs [Día+12], is to answer
the question on the parameterized complexity of Metric Dimension on
planar graphs with respect to the solution size. For this purpose, the key ideas
in our reduction seem to be of limited use. However, there seems to be some
hope that they may help to answer questions on the parameterized complexity
of problems like Identifying Code and variations where, similarly to
Metric Dimension, the basic task is to select a dominating set such
that all vertices are “separated” by their neighborhood to this set (see
Foucaud [Fou12] for a recent thesis on identifying codes). Admittedly, the
most appealing open questions are whether Metric Dimension is fixedparameter tractable with respect to structural graph parameters such as
treewidth and feedback vertex set number and thus we would rather look
at structural parameterizations in future studies. This is mainly because
we consider the study of special graph classes and showing hardness or
tractability with respect to them only as the first step on the way to answer
the question which structures are “responsible” for the intractability of a
problem. For example, having shown that the original W[2]-hardness proof
for Metric Dimension on maximum degree graphs [HN13a] can be extended
185
to hold even on bipartite graphs with maximum degree three leads to the
insight that, putting it in exaggerated terms, odd-length cycles are not the
reason for the computationally intractability of Metric Dimension. Hence,
as Metric Dimension is polynomial-time solvable on trees, one could say
that the existence of any cycles is the reason for its computational hardness,
an observation that seems obvious for many graph problems. Hence, the
need for a more fine-grained analysis naturally arises. Because of this, it is
a natural second step to examine the problem complexity dependence on a
parameter measuring the number of occurrences of a certain structure and
with respect to distance from triviality parameterizations. For example, the
parameters feedback vertex/edge set are an appealing measurement on how
many “non-overlapping” cycles exist. Hence investigating the parameterized
complexity with respect to these parameters can be viewed as a more
fine-grained approach to identify structures which are responsible for the
computational hardness of Metric Dimension.
In Chapter 4 we studied the parameterized complexity of 2-Club with
respect to the structural parameter space of the input graph. To this end,
we arranged the parameters with respect to the “stronger” parameterization
relation which allows to infer consequences of certain (in-)tractability results
from one parameter to another. We showed that 2-Club is NP-hard even on
sparse graphs (parameters average degree and degeneracy) and that even with
respect to the parameter h-index the problem is W[1]-hard. On the positive
side, we have shown that there are polynomial kernels for the rather weak
parameters cluster editing and feedback edge set and that for a large amount
of parameters admitting parameterized algorithms (parameters distance to
cograph and treewidth are the strongest among them) the running time
k
characteristic 2O(2 ) · nO(1) seems to form a challenging barrier. We have
also shown that, unless the SETH fails, the already known parameterized
0
algorithm for the dual parameter k 0 , which runs in 2k · nO(1) time, cannot
be improved. Somehow in contrast to this theoretical lower bound, an
implementation of this parameterized algorithm in combination with a wellknown Turing kernelization and further data reductions rules yield the
currently fastest solver for 2-Club on random data as well as on real-world
instances from social network analysis. Our theoretical findings so far cannot
explain the dramatic success of this rather simple search-tree based algorithm
compared to a general but highly-optimized integer linear programming solver.
The corresponding experimental results demonstrate that there still seem
to be further “hidden structures” in the considered data—further research
is necessary to identify them. In this direction, it seems to be promising
to apply a “data-driven” search for valuable parameterizations. Therein,
the basic idea is to assist the search for promising parameterizations by a
186
systematic and potentially program-conducted search within experimental
data (see for example the Graphana 2.0 program [Grapha 2.0] which assists
in measuring most of the parameters shown in Figures 2.1 & 4.2 on large
graphs). Finally, although some of our results for 2-Club even hold for the
general s-Club problem, it is a natural concern to perform a similar study
of structural parameterizations for 3-Club. 2-Club and 3-Club seem to
be the most relevant cases in practical settings [BBT05, Pas08].
Following the third approach to identify tractable cases of NP-hard problems, we further studied the consequences of different neighborhood structures
on a local search variant of the famous TSP problem (Chapter 5). Our
work was mainly motivated by the aim to analyze the W[1]-hardness proof
of Marx [Mar08] for LocalTSP(Edge) and to determine which properties
of the Edge neighborhood allow for this hardness. We have established a
parameter space of neighborhood structures and based on this we identified
Swap distance as the “weakest” or most restrictive neighborhood. We then
proved that the W[1]-hardness reduction of Marx [Mar08] can be modified
such that it holds even for the Swap distance, implying W[1]-hardness for all
other neighborhood measures under consideration. By the same reduction
we proved that an O(no(k/ log k) )-time algorithm cannot exist (unless the
ETH fails) and this almost closes the gap to the known algorithms running
in O(nk ). We further showed that for the Swap and Edit distances the
permissive local search problem becomes fixed-parameter tractable on planar
graphs. In this sense we showed how the approach to consider special graph
classes can be naturally combined with the local search paradigm in order
to identify tractable cases. Although we demonstrated that the permissive
local search problem on general graphs for the Edge distance is equivalent
to the Reversal distance, we left open the central open question whether
LocalTSP is fixed-parameter tractable on planar graphs with respect to
the size of an Edge neighborhood.
Having shown the usefulness of the three approaches on three different
graph problems, we applied the most promising one, structural parameterizations, to a fourth problem, namely Vector (Positive) Explanation. This
was also motivated by the fact that the standard parameterization by the
solution size is basically (up to the input-length of large numbers) equivalent
to the input size and thus seems to provide limited additional insights helping
to understand the complexity of Vector (Positive) Explanation. We
performed a systematic classification of the parameterized complexity of
the structural parameter space of the input vector. More specifically, “improving” a previous parameterized algorithm with respect to the maximum
value [Bie+13] we proved that the stronger parameter maximum distance
between consecutive entries also admits a parameterized algorithm in case
of Vector Explanation. In addition, motivated by a data reduction
187
rule that relates the solution size k to the number of vector entries n such
that k < n < 2k, we studied the distance from triviality parameterizations
2k − n and n − k. We showed NP-hardness for parameter n − k for both
problem variants and, interestingly, proved that Vector Explanation
is fixed-parameter tractable with respect to 2k − n and, in difference to
that, Vector Positive Explanation is W[1]-hard. Our most challenging
open question is whether the fixed-parameter tractability result for Vector
Explanation with respect to parameter maximum difference in consecutive
vector entries can be transfered to Vector Positive Explanation. Moreover, it would be interesting to see whether the (2k − 1)-entry kernel can be
extended to a “full” kernel, in other words, to find out whether there is a
way to bound the maximum value occurring in the input vector. Finally, it
is a challenging research task to examine the parameterized complexity of
the more general matrix variant.
What all three approaches to identify tractable cases of NP-hard problems
do have in common is that the corresponding parameter spaces admit a
relation that allow to infer (in-)tractability between “weaker” and “stronger”
parameterizations. We demonstrated the usefulness of the three approaches
to systematically chart the border of intractability and to reveal those
structures that determine the computational complexity of a problem. We
believe that studying parameter spaces will turn into a standard method to
analyze NP-hard problems and will establish a natural “race” in parameterized
algorithmics. It turned out that the approach of structural parameterizations
is one of the most promising approaches that waits for being applied to
several problems due to its generic usability. In our opinion it is the natural
next step following the central mission of parameterized algorithmics on the
way to a more fine-grained computational complexity analysis.
We conclude with some general questions and proposals how the study of
parameter spaces should be combined with or extended by other parameterized complexity “tools”.
We have discussed that the parameter space of special graph classes and
structural parameterization are quite related and, indeed, tractability results
on special graph classes are a strong motivation for structural parameterizations. It would be interesting to see whether there are other parameter
spaces, such as solution-oriented parameter spaces related to local search
or structural “solution-parameterizations”, that are related to each other
in a way that inspires and poses interesting new questions. In general, we
believe that the idea of parameterizing the solution search space, where
local search seems to be one framework to study, still waits for a more
extensive exploration.
We would be further interested in whether polynomial-time data reduction
rules can develop into a handy tool to obtain problem-specific parameter
188
spaces. For instance, we have described how a data reduction rule for
Metric Dimension [ELW12] relates the structural parameter feedback
edge set to the solution size parameter. It would be interesting to discover
parameter relations that can be established by applying problem-specific
polynomial-time preprocessing. Can data reduction rules help to reveal
interesting problem-specific parameter spaces?
Multivariate algorithmics [FJR13, Nie10], (roughly) the study of multiple
combined parameterizations, developed into a standard and fruitful technique
in the parameterized “toolbox”. It seems challenging to combine it in a clever
way with the study of parameter spaces, meaning that there is the need
for a systematic way to find reasonable parameter combinations within
parameter spaces.
Last but not least, we believe that following the way to inspect more closely
critical structures causing the intractability of a problem increases the need
for validating theoretical findings in practical settings. We gave a concrete
example where an implementation of a combination of parameterized tools
led to a competitive solving algorithm that can deal with large real-world
data. However, as theoretical insights are, so far, not able to explain this
success, we think that it needs a more interlinked process between parameterized algorithmics and the insights we get from experiments with algorithm
implementations applied to real-world data. In the light of data-driven search
for valuable parameterizations we think that experimental insights would
help to find the right (practically important) theoretical questions. A more
concrete proposal in this direction is to explore the potential of hybrid parameterized algorithms, that is, combining a set of parameterized algorithms
for a problem to an algorithm which chooses, depending on the concrete
parameter values of an instance, the most promising of the algorithms. In
a first step towards this, the Graphana [Grapha 2.0] project pursues the
goal of providing a robust framework automatically measuring structural
graph parameters.
189
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