Applying Convex Integer Programming: Sum
Multicoloring on Bounded Neighborhood
Diversity∗†
Tomáš Gavenčiak1 , Dušan Knop1 , and Martin Koutecký1
1
Department of Applied Mathematics, Charles University
Malostranské nám. 25, Prague {gavento, knop, koutecky}@kam.mff.cuni.cz
Abstract
In the past 30 years, results regarding special classes of integer linear (and, more generally,
convex) programs flourished. Applications in the field of parameterized complexity were called
for and the call has been answered, demonstrating the importance of connecting the two fields.
The classical result due to Lenstra states that solving Integer Linear Programming in fixed
dimension is polynomial. Later, Khachiyan and Porkolab has extended this result to optimizing
a quasiconvex function over a convex set. While applications of the former result have been
known for over 10 years, it seems the latter result has not been applied yet in the parameterized
setting. We give one such application. Moreover, we provide a comprehensive overview of the
state of the art in convex integer optimization from the perspective of parameterized complexity.
Specifically, we deal with the Sum Coloring problem and a generalization thereof called
Sum-Total Multicoloring, which is similar to the preemptive Sum Multicoloring problem. In Sum Coloring, we are given a graph G = (V, E) and the goal is to find a proper coloring
P
c : V → N minimizing v∈V c(v). Using convex integer programming, we show fixed-parameter
tractability results for these problems when parameterized by the neighborhood diversity of G,
a parameter generalizing the vertex cover number of G.
1998 ACM Subject Classification G.2.2 Graph algorithms
Keywords and phrases graph coloring, parameterized complexity, integer linear programming,
integer convex programming
Digital Object Identifier 10.4230/LIPIcs.xxx.yyy.p
1
Introduction
In 1983 Lenstra proved that solving integer linear programming (ILP) is polynomial in fixed
dimension [24]. In the subsequent years this result has been extended by many authors
to wider optimization classes (e.g. minimizing a quasiconvex function over a convex set)
and improved in running time and space. The setting of these results fits well with that
of parameterized complexity. There, the complexity of a problem is studied as a function
of not just the size of the input, but of multiple parameters instead. In the case of convex
integer programming the dimension is a natural parameter.
Consequently, several authors called for applications of ILP in the field of parameterized
complexity, e.g. Niedermeier in his book [30]:
∗
†
Research supported by GAUK grant project 1784214, by the project SVV–2015–260223 and by the
project Kontakt LH12095.
Third author is supported by the project 14-10003S of GA ČR.
© Tomáš Gavenčiak, Dušan Knop, Martin Koutecký;
licensed under Creative Commons License CC-BY
Conference title on which this volume is based on.
Editors: Billy Editor and Bill Editors; pp. 1–13
Leibniz International Proceedings in Informatics
Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany
2
Applying Convex Integer Programming
[...] It remains to investigate further examples besides Closest String where the
described ILP approach turns out to be applicable. More generally, it would be interesting to discover more connections between fixed-parameter algorithms and (integer)
linear programming.
This call has been answered in the following years, most notably by Fellows et al. [8]
in a paper studying graph layout problems parameterized by vertex cover, and by others
studying a related parameter called neighborhood diversity [23, 12].
Curiously, it seemed that a problem can either be tackled already by use of integer
linear programming, or not even the advanced results regarding integer convex programming
suffice. This naturally lead to an interest in problems requiring the use of these advanced
techniques, as Lokshtanov states in his PhD thesis [25]:
It would be interesting to see if these even more general results can be useful for
showing problems fixed parameter tractable.
In this paper, we give one such example. We show that the Sum-Total Multicoloring
problem, a generalization of Sum Coloring, is fixed-parameter tractable with respect to
the neighborhood diversity of the input graph and the number of different vertex demands.
We do so by minimizing a (non-linear) convex function over an integer linear program. We
also discuss how under certain assumptions about a Sum-Total Multicoloring instance
it can be reformulated as a mixed integer linear program. The technique we use to do so
has been known for a long time, but to the best of our knowledge has never been used in
the context of parameterized complexity.
Furthermore, we have collected and summarized advances in the field of convex integer
programming from the past 15 years that we find particularly relevant. We do this with
the hope of providing an accessible toolbox to the benefit of parameterized complexity
researchers.
1.1
Neighborhood Diversity
Courcelle’s theorem [3] from 1990 states that deciding a Monadic Second Order (MSO)
formula on a graph of treewidth k can be done in time f (k, ϕ)n for some computable
function f . However, this function f is non-elementary and this cannot be improved unless
P=NP [10]. Thus a question arose: is there a graph class such that MSO formulas can
be decided in f (k, ϕ)n time for an elementary function f ? This question was answered in
the affirmative in 2012 by Lampis [23] for graphs of bounded vertex cover and neighborhood
diversity.
I Definition 1. The neighborhood diversity of a graph G, denoted by nd(G), is the minimum
number t of classes (called types) of a partition V1 , . . . , Vt of the vertex set VG such that
each type Vi induces either an empty subgraph (independent type) or a complete subgraph (clique type) of G, and
for each distinct Vi and Vj there are either no edges between Vi and Vj , or every vertex
of Vi is adjacent to all vertices of Vj .
Observe in Figure 1 that neighborhood diversity occupies an important place with respect
to other relevant parameters (vertex cover number, treewidth and cliquewidth). Studying
the parameterized complexity of a problem with respect to treewidth can be seen as a
stepping stone to understanding its parameterized complexity with respect to cliquewidth.
T. Gavenčiak, D. Knop, M. Koutecký
cw
mw
tw
Figure 1 A map of assumed parameters. Full arrow stands for
linear upper bounds, while dashed arrow stands for exponential
upper bounds. Note that treewidth and neighborhood diversity
are incomparable. This can be observed by checking that,
nd
vc
3
tw(Kn ) = n − 1
nd(Kn ) = 1
tw(Pn ) = 1
nd(Pn ) = n,
where Kn and Pn are the complete graph and path on n vertices,
respectively.
Since neighborhood diversity is incomparable to treewidth in this regard, it provides a different step towards cliquewidth and gives different insights. The next possible step towards
cliquewidth is so-called modular width introduced by Gajarský et al. [11] as a direct generalization of neighborhood diversity.
Notice that an n-vertex graph of bounded neighborhood diversity can be described in a
compressed way using only O(log n nd(G)2 ) space:
I Definition 2. The type graph T (G) of a graph G is a graph on nd(G) vertices {t1 , . . . , tnd(G) },
where each ti is assigned weight s(ti ) = |Vi |, and where (ti , tj ) is an edge or a loop in T (G)
if and only if two distinct vertices of Vi and Vj are adjacent.
1.2
The Sum-Total Multicoloring Problem
A (proper) coloring of a graph G = (V, E) is a function c : V → N satisfying that c(u) 6= c(v)
whenever {u, v} ∈ E. The cost of c is then the number different colors used, that is |c[V ]|.
The chromatic number χ(G) of G is then the minimum attainable cost of a proper coloring
c of G.
The motivation for this particular formulation is the following coloring problem with a
different objective.
I Problem 3 (Sum Coloring).
INPUT:
TASK:
Graph G.
P
Find a proper coloring c of G minimizing
c(v).
v∈V
In addition to Sum Coloring, we consider a generalized problem combining Sum Coloring with multicoloring. A (proper) multicoloring of a graph G with given vertex demands
(weights) w(v) is a map c : V → 2N such that c(u) ∩ c(v) = ∅ whenever {u, v} ∈ E and
|c(v)| ≥ w(v). Similarly, the objective is to minimize the number of different colors used
S
| c[V ]|. The generalization is then defined as follows.
I Problem 4 (Sum-Total Multicoloring).
INPUT:
Graph G, vertex demands w : V → N.
TASK:
Find a proper multicoloring c of G with demands w minimizing
P
v∈V
P
x .
x∈c(v)
While this generalization is natural, it differs from the well-known Sum Multicoloring
P
problem whose objective function is v∈V maxx∈c(v) x.
When the demands of all the vertices within every single type ti are the same, the
problem is an instance of Uniform Sum-Total Multicoloring. In this case we denote
this demand of the vertices of a type ti by w(ti ). Note that Sum Coloring is a special
case of Uniform Sum-Total Multicoloring with w ≡ 1
4
Applying Convex Integer Programming
The Sum Coloring problem has been introduced by Kubicka and Schwenk [22]. They
showed that a polynomial algorithm for Sum Coloring would be easily modified to compute
the chromatic number and presented a linear algorithm for computing the Sum Coloring
for trees. Furthermore Erdös et al. [7] gave a constructive proof of the unexpected property
that, for any k ≥ 2 and any positive integer t, there exists a family of k-chromatic graphs for
which any optimal sum coloring must use at least k + t colors. This builds upon a previous
result [22], where the same was proven for trees.
The Sum Coloring problem is studied because of the natural connection to scheduling.
Scheduling problems forms a fundamental class of problems studied in the approximation
algorithms theory.
Nicoloso et al. [29] and Halldórsson et al. [13] study the Sum Coloring problem on the
class of interval graphs, together with application to scheduling of dependent jobs. Marx [26]
gave a short proof of NP-completeness of Sum Coloring for interval graphs.
1.3
Our results
We first extend the original idea of Lampis [23] for coloring of bounded neighborhood diversity graphs to multicoloring, and then show how to express the objective function of
Sum-Total Multicoloring, and that it is convex, in the appropriate space:
I Theorem 5. Given a graph G with nd(G) = k and a Sum-Total Multicoloring
instance with d different demands, Sum-Total Multicoloring can be expressed as a
convex integer program in at most (2d)k variables.
Using one of existing algorithms for solving convex integer programming in fixed dimension, we get an FPT result as a corollary:
I Corollary 6. The Sum-Total Multicoloring problem parameterized by nd(G) and the
number of different demands d is fixed parameter tractable.
Moreover, we show that in the special case when the vertex demands are bounded by
poly(n) (by f being poly(n) we mean that there exists c ∈ N such that f ∈ O(nc )), the convex
integer program can be reformulated as a mixed integer linear program in p = (2d)k integer
variables, poly(n) continuous variables and poly(n) inequalities. This, unlike in Corollary 6,
allows applying MILP solvers which are mature and run in space poly(p).
2
Convex Integer Programming (CIP) and Parameterized Complexity
In this section we overview existing results in convex integer minimization (Subsection 2.1)
and maximization (Subsection 2.2) in fixed dimension, and integer linear programming in
variable dimension (Subsection 2.3). The outline structurally follows Chapter 15 of the book
50 Years of Integer Programming [16] omitting some parts but including several more recent
developments.
2.1
Convex Integer Minimization in Fixed Dimension
Lenstra’s result from 1983 shows that solving mixed integer linear programming (MILP)
is polynomial when the integer dimension is fixed [24]. This was significantly improved by
Kannan [20] and Frank and Tardos [9] in two ways. First, the required space was reduced
from exponential to polynomial in the dimension, and second, running time dependency
O(p)
on the dimension p was reduced from 22
to pO(p) . To solve the optimization variant of
T. Gavenčiak, D. Knop, M. Koutecký
MILP using these algorithms one has to do binary search, as described by Fellows et al. [8].
We would like to point out that while Lenstra’s result is old, to the best of our knowledge
mixed integer linear programs were not considered in the parameterized setting before; only
ILPs. However, as we show in Subsection 3.3, under certain conditions MILP can be used
to express a convex objective function in a way which ILP does not allow.
I Definition 7 (partially in Fellows et al. [8]). p-Variable Mixed Integer Linear Programming Optimization (p-Opt-MILP): Let matrices A ∈ Zm×(p+n) , b ∈ Zm×1 and
c ∈ Z1×(p+n) be given. The goal is to find a vector x ∈ Zp×1 × Rn×1 that minimizes (or
maximizes) the objective function c · x and satisfies the m inequalities, that is, A · x ≥ b.
I Theorem 8 (Frank and Tardos [9], Fellows et al. [8]). p-Opt-MILP can be solved using
O(p2.5p poly(L)) arithmetic operations and space polynomial in L, where L is the number of
bits in the input.
This result was later generalized to minimizing a quasiconvex function over a convex
set. A function f : Rn → R is called quasiconvex if for every α ∈ R, the lower set {x ∈
Rn : f (x) ≤ α} is a convex subset of Rn . The first one to show this was Khachiyan and
Porkolab [21], followed by multiple authors. Unlike above, all of the following results require
space exponential in the dimension.
The subsequent research diverged in several directions. The main difference between
the papers we surveyed is in the formalism chosen to represent the convex set. Since there
is no better or worse formalism, choosing one is a matter of preference with respect to a
specific scenario. Another difference is in their motivation – some authors seek to achieve
better time complexity while others contribute by simplifying existing proofs. Our list is
categorized according to the formalism chosen to represent the convex set.
Semialgebraic convex set.
Khachiyan and Porkolab [21] state their result for minimizing a quasiconvex function over
a semialgebraic convex set. These are closely related to spectrahedra, the solution spaces of
semidefinite programs. Independently, convex sets and semialgebraic sets have been studied
for a long time, but together they have been studied only in the past ten years as Convex
Algebraic Geometry. There is a book on the topic by Blekherman, Parillo and Thomas [1]. A
drawback of this result is an exponential dependence on the number of polynomials defining
the semialgebraic convex set.
I Theorem 9 (Khachiyan and Porkolab [21]). Minimizing a quasiconvex function over a
semialgebraic convex set defined by k polynomials in dimension p is FPT with respect to k
and p.
Quasiconvex polynomials. Heinz [15] studied a more specific case of minimizing a
quasiconvex polynomial over a convex set given by a system of quasiconvex polynomials.
His result improves over Khachiyan and Porkolab in terms of time complexity, dropping the
exponential dependence on the number of polynomials. The dependence on the dimension p
3
is O(2p ), which was further improved by Hildebrand and Köppe [18] to O(2O(p log p) ). The
latter result can be stated as follows:
I Definition 10. p-Variable Polynomial Integer Programming Minimization (pMin-PIP): Let quasiconvex polynomials F̂ , F1 , . . . , Fm ∈ Z[x] = Z[x1 , . . . , xp ] with integer
coefficients be given. We want to find a vector x ∈ Zp×1 that minimizes the objective
function F̂ (x) and satisfies the m inequalities, that is, Fi (x) < 0, for all i.
5
6
Applying Convex Integer Programming
I Theorem 11 (Hildebrand and Köppe [18]). Given a p-Min-PIP instance such that d ≥ 2
is the upper bound on the degree of each F , M is the maximum number of monomials in
each F and l bounds the binary length of the coefficients of F , it can be solved in time:
m(rlM d)O(1) 22p log2 (p)+O(p) , thus FPT with respect to the dimension p, if the feasible
region is bounded such that r is the binary encoding length of that bound with r ≤ ldO(p) ,
mlO(1) dO(p) 22p log2 (p) , thus FPT with respect to the dimension p and the maximum degree
d, if the feasible region is unbounded.
Note that, in particular, the running time is polynomial with respect to the number of
polynomials m.
Oracles. Further research lead to splitting the CIP problem in two independent parts
to allow more focus on each of them. The first part is showing that a certain problem
formulation (such as quasiconvex polynomial inequalities, semialgebraic set etc.) can be
used to give a set of geometric oracles. The second part is to show that, given these oracles,
solving a CIP problem can be done in a certain time.
This approach is taken by Dadush, Peikert and Vempala [5] who further improve the
time complexity of Hildebrand and Köppe [18] when the convex set is given by three oracles:
a so-called weak membership, strong separation and weak distance oracles. Observe that
the running time of Theorem 11 can be rewritten as O∗ (p2p ); Dadush et al. improve it to
4
O∗ (p 3 p ). Moreover, Dadush claims in his PhD thesis [4] an O∗ (pp ) algorithm. (Here and in
the following we use the O∗ notation which suppresses polynomial factors.)
This sequence of results can be seen as a part of a race for the best running time.
Dadush classifies existing algorithms as Lenstra-type and Kannan-type, depending on the
space decomposition they use (hyperplane and subspace, respectively). The type of algorithm determines the best possible running time – Lenstra-type algorithms depend on a
so-called flatness theorem, which gives a lower-bound O∗ (pp ). The best known Lenstra-type
4
algorithm is the O∗ (p 3 p ) algorithm of Dadush et al. [5]. Note that both Theorem 9 and 11
are Lenstra-type algorithms. On the other hand, Kannan-type algorithms could run as fast
as O∗ ((log p)p ) if a certain conjecture of Kannan and Lovász holds (Theorem 7.1.3 [4]).
The O∗ (pp ) algorithm given in Dadush’s thesis [4] is Kannan-type. It is also worth noting
that the only known lower bound for CIP in general is the trivial one of O∗ (2p ).
The oracle approach is also taken by Oertel, Wagner and Weismantel [32]. They show
that a general convex integer program given by a so-called first order evaluation oracle
can be reduced to several p-Opt-MILP subproblems, which are readily solved by existing
solvers (implementing for example Theorem 8). In a previous paper [31] the same authors
take a more generic approach requiring a set of oracles to solve a minimization problem, and
discuss how to construct these oracles specifically for the p-Min-PIP problem.
2.2
Convex Integer Maximization in Fixed Dimension
When we make the step from a linear to a general quasiconvex objective function, we have to
distinguish carefully between minimization and maximization. Here we mention two results
that can be applied in this case.
Vertex Enumeration. Under certain conditions, the number of vertices of a given
polytope is small:
I Theorem 12 (Cook et al. [2]). Let P = {x ∈ Rp : Ax ≤ b} be a rational polyhedron with
A ∈ Qm×p and let φ be the largest binary encoding size of any of the rows of the system
Ax ≤ b. Let P I = conv(P ∩ Zp ) be the integer hull of P . Then the number of vertices of
P I is at most 2mp (6p2 φ)p−1 .
T. Gavenčiak, D. Knop, M. Koutecký
Since Hartmann [14] also gave an algorithm for enumerating all the vertices running in
polynomial time in fixed dimension, it is possible to evaluate the convex objective function
on each of them and pick the best. Due to the upper bound on the number of vertices we
see that this gives a simple O∗ (np ) (XP class) algorithm. Moreover, in several cases where
ILP proved to be a successful technique, φ = log n (holds for two out of four problems in
Fellows et al. [8], and all in Lampis [23] and Ganian [12]). Since (log n)k for fixed k is order
o(n), this indicates that solving convex integer maximization might be FPT in many cases.
Reduction to Mixed Integer Linear Programming. A different possibility is reducing a given CIP to a MILP, allowing maximization. Obviously, this can not be done in
every case, but Michaels and Weismantel [28] show a surprisingly large class of CIPs where
this is possible.
We would like to point out here that this approach is relevant even in the case of convex
integer minimization (discussed before). While it does not lead to new time complexity
results, it has two advantages. First, it allows the use of the algorithm from Theorem 8
and thus leads to a reduction in space complexity, and second, it allows exploiting existing
powerful ILP solvers, which are more mature than current CIP solvers (cf. Subsection
15.3.3 [16]).
For the sake of simplicity we state this result in a weaker version. We say a polynomial
F is (additively) separable if each of its monomials is univariate.
I Theorem 13 (follows from Theorem 2 of Michaels and Weismantel [28]). Given a problem of
the form opt{f (x) : Ax = b, A ∈ Zm×p x ∈ Zp×1
+ } where f is a separable convex polynomial
with domain [0, B]p , an equivalent p-Min-MILP instance in p integer variables, poly(m, B)
continuous variables and with poly(m, B) inequalities can be constructed in time poly(m, B).
We discuss how to apply this technique in one specific case in Subsection 3.3, hopefully
providing further insight into its main idea.
2.3
Integer Linear Programming in Variable Dimension
Two major well-known cases of linear programs (LPs) that can be solved integrally in polynomial time are LPs in fixed dimension (as discussed above) and LPs given by totally
unimodular matrices (such as flow polytopes). Here we overview two more cases.
n-fold Integer Programming. A large case introduced and studied in the last 15 years
are so-called n-fold integer programs. It was shown that several important problems can be
expressed as n-fold integer programs, and that n-fold integer programming can be solved
in time O(ng(A) L), where L is the length of the input, and g(A) is the so-called Graver
complexity of matrix A. For an introduction into this theory see a paper of Onn [33].
Observe that if fixing some parameters leads to fixed g(A), we get an XP algorithm. The
algorithm for solving n-fold integer programming was later improved by Hemmecke, Onn and
Romanchuk [17] to O(n3 L), where L depends on g(A), leading to FPT algorithms if g(A)
is fixed. Specifically, the authors use this to show that a multicommodity transportation
problem with l commodities, m suppliers and n customers is FPT with respect to m and l.
The relevance of these results is highlighted by Downey and Fellows [6]:
Conceivably, [ Minimum Linear Arrangement] might also be approached by the
recent (and deep) FPT results of Hemmecke, Onn and Romanchuk [17] concerning
nonlinear optimization.
Bounded Treewidth ILP and beyond. Given an ILP instance A, one can examine
the structure of its underlying Gaifman graph G(A). G(A) is the graph whose vertices
7
8
Applying Convex Integer Programming
correspond to the variables of the ILP, and two vertices are adjacent if the corresponding
variables occur together in an inequality. This way, the graph structural concept of treewidth
can be generalized to ILPs. It can be observed using Courcelle’s theorem that if the size of
the domain of each variable is bounded by d and the treewidth of G(A) is bounded by k, A
can be solved in time O∗ (dk ).
Recently, Jansen and Kratsch [19] introduced so-called r-boundaried ILPs which significantly generalize totally unimodular ILPs and ILPs of bounded treewidth, and give an FPT
result regarding r-boundaried ILPs.
3
3.1
Solving Sum-Total Multicoloring on bounded neighborhood
diversity
Overview
The algorithms of this section are based on the concept of vertex coloring polytope of a graph
(in particular its independent set formulation [27]), where there is a {0, 1}-variable xI for
P
every independent set I of G with the constraint I3v xI = 1 for every v ∈ V . This encodes
which independent sets correspond to actual color classes in a coloring, every vertex being
P
colored exactly once. The goal is then to minimize I independent xI – the number of used
colors.
As previously used by Lampis [23], observe that any independent type ti can be colored
with one color and every clique type tj needs exactly s(tj ) colors. Every color class of such
coloring (and even multicoloring) then corresponds to an independent set in T (G). Thus,
every such coloring can be represented by stating how many times is each independent set
of T (G) used (up to permuting the colors, and permuting the vertices of clique types).
Denote I = {I : I independent in T (G)} the set of independent sets of T (G) and recall that
s(ti ) = |Vi | for every type.
For graphs of small neighborhood diversity and uniform demands, this can be expressed
as the integer linear programs below, which have a variable xI for every I ∈ I. Note that
since multiple color classes of G can correspond to one I ∈ I, xI is no longer a {0, 1}-variable.
In the independent set formulation of the vertex coloring polytope an integer point x
corresponds to a certain coloring up to permuting the colors. In our formulation we only
capture (multi)colorings which, for a given color γ and an independent type ti , either color
all vertices of ti with γ or none of them. We call such colorings essential, and, accordingly,
the polytope corresponding to our formulation the essential (multi)coloring polytope.
In case of non-uniform demands of only few different values we may use the following
transformation.
I Lemma 14. For a graph G with nd(G) = k, type graph T and non-uniform demands w,
P
there is a type graph T 0 with uniform demands w0 such that |T 0 | = ti ∈T |w[Vi ]| where the
last term is the number of different demand values for Vi .
Proof. Take T and replace every vertex (type) ti with |w[Vi ]| = zi clones (vertices with
the same neighborhood), one for every value of w[Vi ], setting w0 to this value and s0 to the
number of vertices with such demand. Then T 0 represents G with equal demands represented
uniformly.
J
While the restriction to a bounded number of different demand values might seem
limiting, bear in mind that, for example, different demand values for every vertex would
generally require Ω(n) sized input, substantially more that the graph itself requiring only
T. Gavenčiak, D. Knop, M. Koutecký
9
O(log n nd(G)2 ) space. This might be an indication that general demands could encode
problems with significantly higher inner complexity.
Essential Coloring and Uniform Multicoloring polytope
Coloring
X
xI = 1
Uniform Multicoloring
X
xI = w(ti )
I∈I,ti ∈I
I∈I,ti ∈I
X
X
xI = s(ti )
I∈I,ti ∈I
for every ti independent
xI = s(ti )w(ti )
for every ti clique
0 ≤ xI
for every ti
I∈I,ti ∈I
0 ≤ xI
Notice that there are at most 2k independent sets in a graph with nd(G) = k. The
Coloring and Multicoloring problems then correspond to the above ILPs minimizing
P
the sum I∈I xI in both cases, counting the different colors used. In essence, these programs
solve a covering problem with I as the covering sets and types ti as the elements to be
covered, requiring each ti to be covered a certain number of times.
Note that together with the results of the previous section, this also shows that Uniform
Multicoloring is FPT with respect to nd(G). However, a stronger FPT result can be
obtained for general Multicoloring by noticing that in any independent type ti only the
maximum demand is relevant in the optimal solution, and similarly, that in any clique type
tj only the sum of the demands matters. We leave the details to the kind reader.
3.2
The Convex Integer Program
While these polytopes also contain all essential solutions to Sum Coloring and SumTotal Multicoloring problems, their objective function is not as straightforward to
obtain. However, given I in a uniform demand setting, the number of vertices every color
class of this type will contain is independent of the actual xI ’s. Namely, define the number
of vertices of a color class σ : I → N ∪ {0} as
σ(I) =
X
X
1+
ti ∈I
ti clique
s(ti ).
ti ∈I
ti indep.
I Lemma 15. Given the numbers xI of occurrences of every independent set I ∈ I as a
color class inside the essential (multi)coloring polytope, the assignment to colors 1, 2, . . .
minimizing the cost of Sum-Total Multicoloring is by decreasing σ(I).
Proof. See Figure 2 for an illustration of the situation. The lemma is shown by looking
at the price difference of swapping the numbers assigned to two color classes with different
σ.
J
To get a quadratic objective function, we express it in terms of auxiliary variables
y1 , . . . , yn which are a linear projection of variables x = (xI1 , xI2 , . . . ). Namely, yj indicates
how many color classes contain at least j vertices:
X
yj =
xI .
σ(I)≥j
Applying Convex Integer Programming
Color number
10
10
9
8
7
6
5
4
3
2
1
x6
x5
x4
x3
x2
x1
y1 y2 y3 y4 y5 y6 y7 y8 y9 y10 y11
yn
Number of vertices in a colorclass (σ)
Figure 2 Illustration of the price decomposition to the individual classes. Note that i-th row
(color i) has price i per vertex.
Finally, the total objective function can be expressed as
S(x) =
p
X
i=1
|iσ(Ii )| =
n X
yj
j=1
2
where i = 1, . . . , p is the order of the color classes given by Lemma 15, every class of type I
present xI times, where we enumerate only those I with xI ≥ 1. The equivalence of the two
is straightforward to check.
Finally, S is convex with respect to x. This is because:
all xI are linear (thus affine) functions,
P
yi = I:σ(I)≥i xI is a sum of affine functions, thus affine,
yi (yi − 1)/2 is convex: it is a basic fact that h(x) = g(f (x)) is convex if f is affine and g
is convex. Here f = yi is affine by the previous point and g = f (f − 1)/2 is convex.
S is the sum of yi (yi − 1)/2, which are convex by the previous point.
We summarize these observations into the following theorem.
I Theorem 16. Given a graph G with nd(G) = k by its type graph T (G) and given uniform
demands w : VT → N, the Sum-Total Multicoloring (as well as Sum Coloring)
problem is FPT with respect to k and can be solved in time O(2k
solver.
O(2k )
(log n)O(1) ) by a PIP
Proof. The essential multicoloring polytope exactly contains all x corresponding to a valid
essential multicoloring. Such polytope has O(2k ) variables and O(k) linear constraints, with
the coefficients represented by O(log n) bits each.
T. Gavenčiak, D. Knop, M. Koutecký
11
zi
6
Figure 3 f (yi ) can be approximated by a piecewise linear
function given by the integer points of f . The first four
resulting inequalities are:
3
zi ≥ 0
zi ≥ yi − 1
zi ≥ 2yi − 3
zi ≥ 3yi − 6
In general, the j-th inequality is zi ≥ jyi − (j + 2)(j + 1)/2.
1
yi
1
2
3
4
The objective function S(x) is a convex function and can be expressed directly in terms
of x by substituting all the yj ’s by their definition. The resulting function can be easily
computed in linear time when the sets of I are preordered by σ.
Finally, the running time follows from the CIP tools presented in the previous section,
e.g. Theorem 11. The Sum Coloring variant is obtained with w ≡ 1.
J
We also get the following by application of Lemma 15.
I Corollary 17. Given a graph G with nd(G) = k by its type graph T (G) and arbitrary
demands w : VG → N, the Sum-Total Multicoloring problem is FPT with respect to k
and the number of different demand values used.
3.3
Reduction to Mixed Integer Linear Programming
As alluded to before (Theorem 13), under certain conditions a p-Min-MILP problem with a
convex polynomial objective function can be reformulated as a pure p-Min-MILP problem.
Here we will show how this can be done in our case if the maximum demand wmax =
maxv∈V w(v) is bounded by poly(n) (in particular for Sum Coloring).
Examine the objective function S(x). It can be seen as an additively separable function
of n convex univariate functions zi = f (yi ) = yi (yi − 1)/2. Equivalently, we could write it
Pn
as S(x) = i=1 zi such that, for every i, zi ≥ f (yi ). Observe that the convex region given
by zi ≥ f (yi ) can be approximated by the piecewise linear function given by all integer
points of f which could be possibly evaluated in the convex integer program, as depicted in
Figure 3.
Formally, our MILP reformulation consists of adding n continuous variables yi as defined
at the beginning of this section, n continuous variables zi as defined above and, for each i,
O(nwmax ) new inequalities in these continuous variables as in Figure 3. The correctness of
this construction follows from the integrality of x, the fact that the yi variables are a linear
projection of x, preserving integrality, and the zi variables are integral due to the integrality
of the yi variables. Obviously, the size of this formulation is poly(n) if and only if wmax is
bounded by poly(n).
12
Applying Convex Integer Programming
4
Conclusion and Open Problems
In Section 2 we have provided a comprehensive overview of existing results in convex integer
programming which are particularly relevant to the field of parameterized complexity. Then,
in Section 3 we have demonstrated how to apply some of these techniques to a particular
problem.
Since our overview in Section 2 is quite extensive, naturally we have not used all tools
that are available. It would be interesting to see more examples of applications of these
tools. In particular, we would like to see an instance of a problem that is solved using
semidefinite programming in fixed dimension, and more instances of applications of n-fold
integer programming.
Regarding specifically the Sum-Total Multicoloring problem there are two natural
directions to explore. First, could we drop the parameterization by the number of different
demands? Observe that then we cannot transform a general instance into a uniform one
(without blowing up the neighborhood diversity), and without that we lose the ordering of
colors by sizes; it is not obvious how to even compute the cost S(x) for a given point x of the
essential multicoloring polytope in polynomial time. Second, could our results be extended
to the larger class of graphs of bounded modular-width? While it is possible for Coloring
by combining ILP and dynamic programming [11], the non-linear objective function of Sum
Coloring makes that approach much more difficult.
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