Graph Packing Problems
Sarah Pantel
B.Sc. (Major), University of Manitoba, 1994
A T H E S I S S U B M I T T E D IN P A R T I A L F U L F I L L - V E N T
O F T H E R E Q U I R E M E N T S FOR THE D E G R E E O F
MASTERO F SCIENCE
in t h e Department
of
Mathematics and Statistics
@ Sarah Pantel 1999
SIMON FRASER UNIVERSITY
October 1999
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Abstract
T h e problem of covering t h e vertices of a given undirected g r a p h with a m a x i m u m number
of disjoint copies of t h e complete g r a p h o n two vertices,
K2,is called t h e m a x i m u m matching
problem. D u e t o t h e fruitful results of matching theory, such as t h e polynomial algorithms
for finding t h e maximum m a t c h i n g of a given g r a p h , it was hypothesized t h a t similar results
might hold for the analogous problem using a g r a p h o t h e r t h a n K 2 . T h e problem of covering
a g r a p h with copies of g r a p h s o t h e r t h a n only K2 is called t h e graph packing problem. Let
G be a family of graphs. Formally, a G-packing of a g r a p h H is a s e t of vertex disjoint
s u b g r a p h s { G 1 , ...,Cik) of H such t h a t each G; is isomorphic t o some g r a p h in <3. T h e
g r a p h packing problem h a s been studied for a variety of families G . Although t h e positive
results a r e n o t a s a b u n d a n t a s for t h e matching problem, m a n y polynomial results have
been found for g r a p h packing problems o n undirected graphs.
The undirected graph packing problem has now been studied quite extensively. However.
this is not t h e case for t h e directed packing problem. In this thesis, we s t u d y t h e directed
g r a p h packing problem for families of directed p a t h s , directed cycles a n d directed stars. T h e
main new result of t h e thesis is a Berge-like characterization of maximum packings for t h e
directed g r a p h packing problem with t h e family
6).
Acknowledgments
Firstly, I would like t o acknowledge my senior supervisor, Dr. Pavoi Hell, for his direction
and advice on this thesis. I would also like t o thank m y other supervisors Dr. Brian AIspach
and Dr. Rick Brewster for serving on t h e committee. I am very grateful t o Dr. Brewster
for his invaluable help with the editing of this thesis, with the proof of the main result of
Chapter 3 as well as for his financial support. I would also like t o thank N.S.E.R.C. for the
financial support of a PGS A award throughout most of this degree and the Department of
Mathematics and Statistics for various TA positions.
I would like t o thank my parents for their encouragement and prayers a n d my parentsin-law for their support.
Lastty, but most importantly, I'd like t o thank my husband.
Christian, for constant encouragement, understanding and hours of proof-reading throughout the preparation of this thesis and the Lord Jesus Christ for helping m e t o persevere
with my studies and for giving me strength and the right perspective.
"Give me, 0 Lord, a steadfast heart, which no unworthy affection may drag
downwards: give me a n unconquered heart, which no tribulation can wear out;
give me an upright heart, which no unworthy purpose may tempt aside. Bestow
on me also, 0 Lord my God, understanding t o know you, diligence t o seek you,
wisdom t o find you, and a faithfulness t h a t may finally embrace you, through
Jesus Christ o u r Lord."
Thomas Aquinas
"I said, 'You a r e my servant'; I have chosen you and have not rejected you. So
d o not fear, for I a m with you. Do not be dismayed, for I a m your God. I will
strengthen you a n d help you; t will uphold you with my righteous right hand."
Isaiah 4 1:9-10 (NIV)
Dedication
To my husband Christian
Contents
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...
111
.-lcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . is
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1
Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3
T h e Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.4
Complexity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Survey of the General Graph Packing Problem . . . . . . . . . . . . . . . . . 9
2.1
Families of Complete Graphs . . . . . . . . . . . . . . . . . . . . . . . 9
2.1.1
{K2,
Fi)-Packings . . . . . . . . . . . . . . . . . . . . . . . . 18
Abstract
2.2
Families of Complete Bipartite Graphs
..................
'LO
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4
Families of Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5
Families of Mixed Graphs . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.6
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Directed Graph Packings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1
Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2
Packing with Directed Paths . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.1
One-Element Families . . . . . . . . . . . . . . . . . . . . . . 3'2
Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
Packing with Directed Cycles . . . . . . . . . . . . . . . . . . . . . . . 59
2.3
Families of P a t h s
3.5
Packing with Directed Stars
........................
62
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -13
-I
Conclusion
vii
List of Tables
. . . . . . . . . . . . 69
4.1
Complexity of t h e packing problem for families of cliques
4.2
Complexity of the packing problem for families of complete bipartite graphs
4.3
.
TO
Complexity of t h e packing problem for families of p a t h s
70
4.1
Complexity of t h e packing problem
TO
4.5
Complexity of t h e packing problem
4.6
Complexity of t h e packing problem
............
for families of cycles . . . . . . . . . . . .
for mixed families . . . . . . . . . . . . .
in t h e directed case . . . . . . . . . . . .
71
71
. . . . . . . . . . . . . . . . . . . . 37
3.6 A COL.{~?,}.factor of the gadget including the connectors . . . . . . . . . . . 37
3.7 A COL-{fi}-factor of the gadget excluding the connectors . . . . . . . . . . . 38
3.8 Placement of fi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
35
Gadget for the COL-{fi}-factor problem
3.9
.........................
X G-Factor including the connectors . . . . . . . . . . . . . . . . . . . . . . .
A G-Factor excluding the connectors . . . . . . . . . . . . . . . . . . . . . . .
Examples of augmenting configurations for the {fi,
problem . . .
...
Augmenting configuration components for the (6.f i ~ - ~ a c l i iproblem
n~
3.10
3-11
3.12
3.13
Gadget for path factor problems
40
41
41
4.5
46
{ f i , fi}-packing problem . . . 47
3.15 Example diagrams for case 2b . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.14 Examples of augmented configurations for the
...........................
Example diagrams for case 3a . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example diagrams for case 3b . . . . . . . . . . . . . . . . . . . . . . . . . . .
Gadget for t h e {c3)-factor problem . . . . . . . . . . . . . . . . . . . . . . .
A {c;}-factor
containing the connector vertices . . . . . . . . . . . . . . . . .
A {c3}-factor excluding the connector vertices . . . . . . . . . . . . . . . . .
Gadget for t h e {c;}-factor problem . . . . . . . . . . . . . . . . . . . . . . .
3.16 Example diagrams for case 2c
54
3.17
57
3.18
3.19
3.20
3.21
3.22
58
60
60
61
62
. . . . . . . . . . . . . . . . . 63
3.24 A {c4}-factor excluding the connector vertices . . . . . . . . . . . . . . . . . 64
3.23 A {c4)-factor containing the connector vertices
-
........................
Gadget for the {S3}-factor problem . . . . . . . . . . . . . . . . . . . . . . . .
3.2.5 Gadget for the {S2)-factor problem
64
3.26
65
................
3.28 .4 n {S3)-factor excluding the connector vertices . . . . . . . . . . . . . . . . .
66
........................
67
3.27 An {&)-factor containing the connector vertices
3.29
Gadget for the (5';)-factor
problem
66
Chapter 1
Introduction
T h e aim of this thesis is two-fold.
W e will survey results o n the general g r a p h packing
problem for undirected graphs a n d t h e n we will begin to develop a similar theory for directed
graphs.
1.1
Definitions and Notation
In order to understand the terminology of t h e g r a p h packing problem, we must first introduce
some definitions from graph theory a n d from matching theory.
A simple graph, o r just gruph, H is a n ordered pair ( V ( H ) ,E ( H ) ) , where V ( H ) =
{vl , . ..,v,)
a n d E ( H ) = ( e l , . . ., e m ) . T h e elements v ; , i = 1 .. .n, of V ( N ) a r e called
vertices a n d t h e elements e;: i = 1 .. .m , of E ( H ) a r e called edges. Each edge e; is a n
unordered pair of vertices. These two vertices a r e incident with t h e edge a n d adjacent to
each other. T h e degree of a vertex is t h e n u m b e r of edges with which i t is incident. T h e
size of t h e vertex-set of a graph, I V ( H ) I , is called t h e order of the graph.
A visual representation of this g r a p h is given in Figure 1.1.
.4 directed graph o r digraph,
D, is
defined by t h e ordered pair ( V ( D ), A ( D ) ) . Here,
CHAPTER 1. INTRODCiCTIOrV
Figure 1.1: H = ( V ( H ) ,E ( H ) )
I f ( D ) ,is the set of vertices of D ,a n d A ( D ) is t h e set of arcs o r directed edges of D. Each
arc is a n ordered pair of vertices. T h u s if e; = vjvk is a n a r c , then t h e directed edge e;
begins in vertex v, and e n d s in vertex vk.
A complete gmph, o r clique, o n n vertices, I<,? is a graph with vertex-set { u l , .. .:u,}
and edge-set {vivj li # j ) . A complete bipartite graph, K,,,, is a graph with vertex-set
{ u l l . . .,V , } U { ~ ~ ,.. .,un} a n d edge-set {viujli E (1,.
bipartite graph
denoted
KL,,is called
( 1 , . . ., n}}. T h e complete
a star o n n edges a n d is denoted S,.
Pk,is a graph with vertex-set
. . .,uk} and
(uo, v ~ ,
e; = V;_~L';. A (u, v)-path is a p a t h such t h a t
I;, denoted
. .. rn). j E
t.0
A path of length k.
edge-set ( e l ,. . .. el;) such t h a t
= u and uk = u. Finally, a cycle of length
Ck,is a graph with vertex-set { Q , . .., v k ) a n d edge-set { e l , . . ., e k ) such t h a t
e; = v;v;+l, i = 1,. ..k
- 1, and
ek
= vkul. An example of each of these graphs is shown in
Figure 1.2.
Figure 1.2: Examples of commonly used gra.phs
W e will now introduce s o m e necessary definitions from matching theory. Consider a
graph H = (V,E ) . A graph I< = (V', E') such t h a t V' C V and E'
of H a n d we write It*C
-
5 E is called
a subgmph
W . A matching M of a graph H is a s e t of pairwise vertex disjoint
C'HA PTER 1. INTRODUCTION
edges of H . -4 vertex of H which is incident with a n edge of hl is cocered by ibl a n d is
said t o be in t h e matching M . T h e o t h e r vertices a r e called uncovered by Ad, o r ezposed in
:1/. A p a t h in H is a subgraph A- of H such t h a t Ii is a path. .An allernatiny pdh in H .
with respect t o a matching 1C.l in H. is a p a t h in H whose edges a r e alternately in and o u t
of t h e matching M . A n augmenting path is a n alternating p a t h which begins a n d ends in
esposed vertices. T h e size of a matching is t h e number of vertices covered by t h e matching.
A matching of largest size is called a mazimum matching. If a matching covers all vertices
of
N ,then
it is a perfect matching o r I-/actor of H . A perfect matching of Cs is shown in
Figure 1.3 where t h e bold edges represent edges in t h e matching.
Figure 1.3: A perfect matching of Cs
In matching theory, we usually search for maximum matchings o r 1-factors of graphs.
A result established by Berge [8]s a y s t h a t a matching M of a g r a p h H is maximum if a n d
only if H contains n o augmenting p a t h with respect t o M . Edmonds uses t h e search for
augmenting p a t h s approach t o find a n algorithm for maximum matchings in general g r a p h s
[20]. A theorem of Hall 1261 for bipartite g r a p h s gives a necessary a n d sufficient condition
for t h e existence of a matching covering each vertex in o n e of t h e parts. Let H be a bipartite
g r a p h with bipartition (X,Y) a n d let N ( S ) denote t h e neighbourhood of t h e s e t S C
X.
T h i s neighbourhood is t h e set of all vertices of H which a r e adjacent t o vertices of S. Hall's
theorem s t a t e s t h a t t h e bipartite g r a p h H contains a matching which covers all vertices of
-Y if a n d only if IN(S)I
> IS1 for all S C X.
A necessary and sufficient condition for t h e existence of a perfect matching o f a n arbitrary
g r a p h was found by T u t t e [45]. Before we give t h i s condition, we introduce s o m e necessary
definitions. A non-empty graph
H
is connected if for all u , u E V there exists a (u, v)-path.
An induced subgmph S of H is a s u b g r a p h of H with vertex set V ( S ) C V ( H ) and edge set
E (S)5 E ( H ) such t h a t E ( S ) is m a d e u p of those edges of E ( H ) incident only with vertices
in t h e vertex set V ( S ) .A subgraph S of H is a component of H if S is connected a n d t h e r e
C'HA PTER I. INTRODUCTION
d o e s n o t exist a n edge ut. where u E 17(S)a n d v E V ( H )\ I'(S).
.A component is odd if
t h e number of vertices in t h e c o m p o n e n t is o d d . Now consider t h e g r a p h H
from H by removing all vertices of t h e s e t S
\ S obtained
C V ( H ) and all edges incident with those
vertices. T u t t e proved t h a t a g r a p h H h a s a perfect matching if a n d only if t h e n u m b e r of
o d d components of H \ S is less t h a n o r equal t o
(SI for all S 2 V ( H ) .
T h e richness of these results from matching theory suggests t h a t expanding t h e theory
m a y also be fruitful. Let
G = (GI . . .,Gk)b e
a family of graphs. .A G-packing of a g r a p h
H is a set of pairwise vertex-disjoint s u b g r a p h s of I1 each of which is isomorphic t o s o m e
g r a p h in t h e family G . .A vertex of I1 is covered by t h e G-packing if i t belongs t o a s u b g r a p h
t h e n we call a G-packing a G-packing. Since there is a n a t u r a i
of t h e G-packing. If G = (G),
correspondence between t h e edges of a g r a p h a n d t h e s e t of s u b g r a p h s isomorphic t o
K2,i t
is clear t h a t every I<*-packing of a g r a p h H corresponds to a matching of H a n d vice versa-
T h e size of a packing is t h e n u m b e r of vertices of
W covered by t h e packing. A G-packing of
H of largest size is called a mazimum G-packing of H. If t h e G-packing covers all vertices
of H , then it is a perfect G - p c k i n g o r a G-factor of H.
1.2
Background
T h e founders of matching theory a r e generally agreed t o be J u l i u s Petersen (1839-1910)
and Denes Konig (1884-1944). Petersen is best known for his results o n regular g r a p h s a n d
Konig for his results o n bipartite graphs. O t h e r mathematicians instrumental in matching
theory were Euler, Kirchhoff a n d Tait. .4 good history of t h e work of Petersen, Konig a n d
their contemporaries c a n b e found in t h e preface of L o v k a n d Plummer's book Matching
Theory [42]. An history of g r a p h theory from 1736 t o 1936 c a n b e found in [lo], a n d a
"Sampler of major events of matching theoryn can b e found in [43].
Matching theory has m a n y interesting applications in a r e a s such as scheduling problems
[27], assignment problems [Xi,331, t r a n s p o r t a t i o n problems a n d network flow problems
[23, 331, t h e shortest p a t h problem [33], t h e travelling salesman problem [13, 2.51, a n d t h e
Chinese Postman problem [21]. Results from packing theory have useful applications to
c o d e optimization [ l l ] , clustering [32], a n d component placing [34]. Hell a n d Kirkpatrick
considered o n e such application in [36] where they write t h a t their interest in t h e s t u d y of
generalized matchings s t e m m e d from t h e s t u d y of scheduling examination periods.
CH.4 P T E R I . IXTROD UCTION
1.3
The Problem
In t h i s thesis, we s t u d y 6-packing problems a n d G-factor problems. We have s t a t e d t h a t
t h e theory of packings is a n extension of matching theory. T h e maximum matching problem
is t h e problem of searching for a matching M of a given i n p u t g r a p h H . such t h a t :\
ccvers
I
a m a s i m u m n u m b e r of vertices. T h e analogous problem in packing theory, t h e maximum
G-packing problem, which we will refer t o simply a s t h e G-packing problem, is t h e problem
of searching for a maximum G-packing of a n input graph H . M a n y polynomial algorithms
e s i s t for solution o f t h e maximum matching problem, [20, 19, 2'2,421, a n d it will b e shown in
C h a p t e r 2 t h a t m a n y polynomial algorithms have also been found for t h e G-packing problem
for various families
G.
In matching theory, we s t u d y a special case of t h e maximum matching problem called
t h e perfect matching problem. T h i s is t h e problem of finding a p e r f s t matching of t h e input
g r a p h H. Formally, a decision problem is a function from t h e instances of a problem t o t h e
answers._-_
"yes" a n d "no". For example, t h e perfect matching problem is formally defined as
a decision problem below:
Perfect M a t c h i n g :
Instance:
A graph H
Question:
Does H a d m i t a perfect matching, t h a t ist is there
a matching of H of size IV ( H )I?
T h e packing problem analogous to t h e perfect matching problem is t h e G-factor problem.
I t c a n b e s t a t e d formally as a decision problem as follows:
Given t h e family of g r a p h s
G-factor:
G?
Instance:
A graph H
Question:
Does H a d m i t a G-factor, t h a t is, is there a packing of H
using g r a p h s from t h e family
6 such t h a t t h e size of t h e
packing is IV(H)I?
In C h a p t e r 2, we survey results for t h e G-packing problem. In C h a p t e r 3, we will s t u d y
t h e 6-factor problem in t h e directed case. T h i s s t u d y is from t h e point of view of complexity
theory, which is introduced in t h e following section.
C'H.4 PTER 1 . INTRODUCTION
Complexity Theory
To begin, we will define three broad complexity classes: t h e class P:the class AeP; t h e class
co - JVP. Complexity theory is a fundamental theory of computing science and as such a
f d 1 introduction is beyond t h e scope of this thesis. T h i s section serves as a n introduction t o
t h e areas of complesity theory needed for this work. For a full introduction t o complesity
theory, t h e reader is referred to p23].
A problem is said t o be in t h e class P if it can b e solved by a polynomial time deterministic algorithm. Polynomial time means t h e number of steps executed by t h e algorithm
is bounded by a poiynomial in t h e size of t h e input. For example, a reasonable measure
for t h e size of a n input g r a p h might b e t h e number of vertices o r t h e number of edges.
Deterministic means t h a t t h e i
+ l S t s t e p of t h e algorithm
is uniquely determined by t h e
first i steps.
T h e class AfP contains decision problems which c a n be solved in polynomial time by
a nondeterministic algorithm. -4 nondeterministic algorithm h a s two stages: t h e guessing
s t a g e a n d t h e checking stage. T h e r e a r e two possible outcomes of a nondeterministic algorithm. Either t h e instance of t h e decision problem is a yes-instance ( t h e answer t o t h e
question is yes), in which case t h e checking s t a g e of t h e algorithm s t o p s a n d answers yes,
o r else t h e instance is n o t a yes-instance, in which case! t h e algorithm may s t o p and answer
no, o r it may never stop. For example, consider a nondeterministic algorithm for t h e perfect
matching problem. Let t h e i n p u t g r a p h H have IV(H)I = n a n d let i = 1,. .., n/2. In t h e
guessing stage, choose a n edge of H a t each of the i steps of t h e algorithm. T h i s gives
4
edges a n d this guess at a solution is called a certificate. T h e certificate must b e polynomial
in size. In t h e checking stage, check whether o r not these
2 edges form a perfect matching.
T h i s p a r t of t h e algorithm m u s t complete in time polynomial in t h e size of t h e input. In o u r
esarnple, determining whether o u r certificate is a perfect matching of H can be achieved in
polynomial time. A yes-certificate of a problem is a certificate which upon testing gives t h e
answer yes to the decision problem question. In o u r example, a yes-certificate is a perfect
matching. A n instance of t h e problem is called a yes-instance if there exists a yes-certificate.
Any decision problem solvable by a polynomial ti me deterministic algorithm is solvable
by a polynomial time nondeterministic algorithm since we can simply use t h e deterministic
algorithm as the checking s t a g e of t h e nondeterministic algorithm with a n e m p t y guess.
Therefore, t h e problems in t h e class
P a r e also in t h e class n/P a n d
so P
SJ ~ P .
T h e class co - jVP contains t h e complernenl problems.
class ,VP. Formally, if
n denotes a decision
problem, then
nC,t o those problems ll in t h e
co - ,tip = {llcln E N P ) . T h e
p r o b l e n ~llc is t h e s a m e problem a ll but with t h e answers reversed. -4s explained in ['L-I].
'-If t h e problem 'Given I , is X t r u e for I?' can b e solved by a polynomial t i m e deterministic
algorithm, then so c a n t h e complementary problem 'Given I , is X false for I?' T h i s is
because a deterministic algorithm halts for all inputs, so all we need t o d o is interchange
t h e 'yes' and 'no' responses". -4 polynomial transformation from one decision problem .4 to
another decision problem
B is a function f from t h e instances a of A
to t h e instances b of R
such t h a t this function c a n be computed in polynomial time a n d for all instances of A, t h e
instance is a yes-instance if a n d only if f (a), is a yes-instance of B. I t is n o t clear t h a t all
problems can b e polynomially transformed t o their complement problems. M a n y problems
N P a n d so i t is conjectured t h a t JVP # co - JVP. It
has also been conjectured t h a t P = NP f~
co - n/Pt but this conjecture remains a n open
in co - jVP d o not seem t o be in
problem.
In this thesis, we will prove A'P-completeness of various packing problems. In order
t o define what is m e a n t by AfP-complete, we must introduce t h e concept of reducing o n e
problem t o another- Consider t h e following decision problems A a n d B:
A:
Instance: a
Question: Is a this way?
B:
Instance: b
Question: Is 6 t h a t way?
We will be using t h e term polynomial time reduction, o r simply reduction in this thesis.
This technique is defined in [24 as follows: "The principal technique used for demonstrating
t h a t two problems a r e related is t h a t of "reducing3 o n e to t h e other, by giving a constructive
transformation t h a t m a p s a n y instance of t h e first problem into a n equivalent instance of
t h e second. Such a transformation provides t h e means for converting a n y algorithm t h a t
solves t h e second problem into a corresponding algorithm for solving t h e first problem.''
Therefore, if we have a polynomial time reduction from problem A to problem B, t h e n a
polynomial-time algorithm for B implies t h a t there is a polynomial-time algorithm for A.
We denote a polynomial time reduction of problem .4 t o problem B by A a O.
Now we can define t h e complexity class of m - c o m p l e t e problems. A problem is AfPcomplete if it is in t h e class NP a n d if all other problems in t h e ciass N P c a n b e polynomially
reduced t o it. T h e foundations of n/P-completeness theory were [aid in [14]. Stephen C o o k
emphasized t h e importance of polynomial time reductions. He gave t h e first proof t h a t t h e
CHAPTER I . IIVTRODUCTION
satisfiability problem (SAT) had t h e property t h a t every o t h e r problem in the class A f P
could be polynomially reduced t o it. T h i s property was found to b e t r u e of a number of
problems a n d became known as ~ffP-complete. T h e class of ArP-complete problems is a
subset of JVP containing t h e hardest problems of t h e class JVP. Considering Cook's result.
in order t o prove JVP-completeness, we d o n o t have t o reduce every problem in iVP t o o u r
problem. Instead, we use t h e following theorem.
Theorem 1.4.1. [24] Let ill and
n2 be problems-
If I l l a 112 and 112 E JVP? then I l l is
,VP-complete implies F12 is n/P-complete.
Thus, to prove t h a t a problem 112 is N P - c o m p l e t e , we first show t h a t it is in JVP
and then s h o w t h a t a known N P - c o m p l e t e problem il 1 c a n b e polynomially reduced t o t h e
problem l12. T h e r e a r e various ways t o d o this reduction in o r d e r t o prove JVP-completeness.
T h e three techniques which a r e most frequently used a r e local replacement, restriction,a n d
component design [24]. In t h i s thesis, we will use t h e local replacement technique.
Finally, a problem ll is in t h e class M P - h a r d if s o m e N P - c o m p l e t e problem can b e
polynornially reduced to it.
Chapter 2
Survey of the General Graph
Packing Problem
T h e G-packing ~ r o b i e r nhas been extensively studied over t h e last few decades.
In this
survey, we will consider results o n the general undirected g r a p h packing problem which asks
"How many vertices of H c a n we cover with copies of g r a p h s in t h e family
G?". in
addition
to t h e general g r a p h packing problem, many o t h e r packing problems have been defined and
studied.
In this chapter, we will s e p a r a t e results according t o t h e t y p e of family
G considered.
We
begin by considering families of complete g r a p h s and then move o n to families of complete
bipartite graphs, of paths, of cycles, and finally to mixed families. T h e results in this chapter
c a n be found in (4,
2.1
.jl 15, 17, 24, 27, 28,
29, 30, 36, 38, 39, 401.
Families of Complete Graphs
Consider t h e graph packing problem for
G = {Kt). This is clearly polynomial
since for a n y
g r a p h , the set of vertices is a perfect {Kt)-packing. T h e G-packing problem for G = {K2)
is also polynomial since this is simply t h e maximum matching problem.
T h e next family to consider is
G = {Ic3}.
T h i s is sometimes referred t o as t h e triangle
packing problem and i t is N P - c o m p l e t e [ l i ] . T h e NP-completeness proof can be done
by reduction of t h e %dimensional matching problem (3DM) t o t h e {Ic3}-factor problem as
shown below. T h e problem 3DM is a n N P - c o m p l e t e decision problem which looks for a
C'HAPTER 2. SURVEP' OF T H E G'ENERAL CZ'RAPH P.-\CIiINC; PROBLEM
3-dimensionat matching of a s e t of triples T . T h i s problem c a n be s t a t e d as follows:
3DM:
Instance:
An integer q. s e t s W. -3-, a n d
a s u b s e t ?-of
Question:
M'x
X
x
).'
with 11371 = 1-X-I = IYI = q. a n d
Y.
Is t h e r e a s u b s e t M of T containing q triples n o t w o of which
use t h e s a m e element from a n y of t h e s e t s W , X- o r Y?
T h e n e x t result is s t a t e d in [29] a n d we give o u r own proof below.
Theorem 2.1.1. The {li3)-factor problem is J\~P-complete.
Proof: T h e {K3)-factor problem is in AfP since given a potential {K3)-factor
of a g r a p h H as a certificate, w e c a n verify in polynomial t i m e t h a t all vertices
of H a r e covered by exactly o n e
K3.W e will reduce t h e AfP-complete problem
3DM t o t h e (IT3}-factor problem t o establish NP-completeness.
Let 7 = ((ti,t * , ts)ltL E W,t 2 E .Y, t3 E
Y ) b e s o m e subset of FV x X x Y where
W ?X, Y = (1,2, . . .,q}. W e shall c o n s t r u c t a g r a p h G such t h a t t h e s e t o f triples
7 has a 3DM if a n d only if G a d m i t s a (lie3)-Factor.
In N P - c o m p l e t e n e s s proofs we will often use t h e local replacement technique.
In t h i s technique, we use gadgets to c o n s t r u c t a n instance of o u r problem. In
t h i s proof we will use t h e g a d g e t shown in Figure 2.1. This g a d g e t is used t o
c o n s t r u c t a g r a p h G.
Figure 2.1: G a d g e t for the {K3)-factor problem
Construction: Consider a g r a p h having vertex s e t W
u X u Y.
We construct
10
t h e edge s e t ,as follows. For each triple (t 1. t2, t3) of T , identify t h e connector
oertices, a . 6 , and c, of a copy of t h e gadget with t h e vertices t l . t 2 a n d t 3 ,
respectively. of t h e graph. All o t h e r vertices, t h e interior vertices, of each copy
of t h e gadget become vertices a n d t h e edges of t h e copies of t h e gadget become
edges of t h e graph. CaIll t h e resulting g r a p h G.
Claim 2.1.1. T h e set 7 has a three dinrensional matching v a n d only if G
a
{K3}-factor.
Proof: Suppose T h a s a 3DM. T h e n for triples ( t l , t 2 ,t3) in t h e matching, t h e gadget with connector vertices a = t l , 6 = t2 a n d c = t3 h a s
t h e {Ii-3)-factor shown in Figure 2.2.
Figure 2.2: A {K3)-factor containing t h e connector vertices
For triples not in t h e matching, t h a t i s when t l , t z , and t3 a r e each
covered by other triples, t h e copy of t h e gadget having vertices t l , t 2
a n d t3 h a s t h e {[GI-factor of Figure 2.3.
Thus, there i s always a (K3)-factor of G when there is a 3DM of 7.
Now, suppose G has a {Ic3)-factor. Consider some copy of t h e gadget.
The point z3 must either b e covered by t h e triangle 21x223 o r else by
t h e triangle y323x3. Suppose zs i s covered by y32323. Then in order to
cover t h e vertex 21, we must choose t h e triangle a z l x z . Similarly, we
m u s t choose czlz2 a n d byly2 a n d we have a {&)-factor
of t h e gadget
as in Figure 2.2. In this case, since t h e connector vertices a r e covered
has
CH.4PTER 2. SURVE)' OF T H E G E X E R AL GRAPH P.~CI\'ILV(:PROBL EILI
Figure 2.3: A {Ic3)-factor excluding the connector vertices
by triangles within t h e gadget, we choose the triple ( a ,6,c ) t o b e in
the 3-dimensional matching of T.
Now, suppose t h a t
23
is covered by the triangle ~
~ ~ 2 Then
x 3 . vertex y3
must b e covered by the triangle yly2g3 and the vertex
~ 1 2 2 ~ In
3.
z3
by the triangle
this case, t h e connectors are not covered by triangles in t h e
gadget a n d so we do not choose (a, 6,c) as a triple of the 3-dimensional
matching.
T h e resulting set of chosen triples T = { ( a . byc ) ) is a 3-dimensional
matching. 0
Since 3DM is a n n/P-complete problem and we have shown that it reduces t o
t h e {Ic3)-factor problem, the (Ii-3)-factor problem is also JVP-complete. CI
Hell a n d Kirkpatrick [35, 361 have proven t h a t the problem of packing a n input graph
K 2 is NP-complete. However, when we consider
packing with a g r a p h G which is the disjoint union of o copies of K Ia n d P copies of Am2
with a connected g r a p h other t h a n K t o r
then t h e G-packing problem is again polynomial. Consider t h e following theorem:
Theorem 2.1.2. [36] Let G be the disjoint union of a copies of I<-iand P copies of K2Then H admits a G-factor if and only if ( V ( G ) (divides ( V ( H ) I and some matching of H
has at least 3!,
edges.
P m f : Suppose H admits a G-factor. If (V(G)Idoes not divide IV(H)I then
we cannot cover all vertices of N with copies of G such t h a t a n y vertex of H is
CH.4PTER 2. SURVEY OF T H E GEiVER4 L GR4 PH PAC'IilNC PROBLEM
13
in exactly o n e copy of G'. Therefore, IV(G)I divides II'(H)I. Within t h e set of
vertices of H used by each copy of G of t h e packing, there needs to be 0 copies
of
K 2since G has this number of K2's. T h i s is true for each of the disjoint copies
of G, of which there a r e
\%I, a n d so in total, t h e number of edges of some
V ( H ) edges. Conversely, suppose IV(G)I
matching of H must have a t least @{V(C)/
divides ]V(H)I and in addition, some matching M of H has at least
edges. T h e n we cover vertices of
H in t h e following way: choose
V(H)
flfm/
edges from
:I#
a n d s o m e a vertices from H \ M a n d let this be a copy of G in the C-packing.
IV(W
These chosen edges and vertices c a n n o t b e used again. Since there a r e fl1t/OIl
edges in t h e matching, we can define
{w}
copies of G in this way. Clearly, the
G-packing is perfect a n d so this defines a G'-factor of H. 0
For g r a p h s G satisfying t h e hypothesis of Theorem 2.1.2, the algorithms used to find a
maximum matching c a n be used t o determine if H admits a G-factor. Since determining
the hypotheses of Theorem 2.1.2 can be d o n e in polynomial time, we have t h e following
corollary:
Corollary 2.1.1. [36] If G is the disjoint union o f 0 copies of ITI and /3 copies of Ka,then
the G-factor problem is polynomial.
It is also proven in 1361 t h a t for all o t h e r g r a p h s G, the G-factor problem is N P - c o m p l e t e .
T h e packing problem with G = {K2,K 3 ) h a s been studied quite extensively, see [ l i . 291:
and h a s been found to be polynomial. Hell a n d Kirkpatrick t r e a t t h e {K2,K3)-packing
problem in C29] a n d prove the following theorem.
Theorem 2.1.3. [29] Let G = {Am2,K3}. Then there exists a polynomial-time algorithm to
find a maximum size G-packing o j arbitrary input gmph H .
In t h e proof of Theorem 2.1.3, the a u t h o r s use Lemma 2.1.1 below which is analogous
t o t h e result from matching theory t h a t a matching of H is maximum if and only if H h a s
no augmenting path. Before we state the lemma, some more definitions are required. First,
consider G a
{K2,
K3)-packing of a graph H. Edges of H which a r e in t h e packing a r e
called G-edges and subgraphs of H in t h e packing which a r e isomorphic t o K3 a r e called
C'HAPTER 2. SURVEY OF THE GENERAL GRAPH PACIiiNG' PROBLEM
Path
***
vo
"1
"0
"1
@
'n-I
vn
14
v
n+ 1
Tail
"2
..
n+3
Kite
Figure 2.4: .4ugmenting configurations for t h e {K2,
K3)-packing problem
G-triangles. An ultemaling path of H with respect t o G is a p a t h in H whose edges a r e
alternately in G a n d n o t in G. T h a t is, if t h e first edge of t h e p a t h is in t h e packing, then
t h e next edge of t h e p a t h is n o t in t h e packing, e t cetera. An augmenling path of H with
respect to G is a n alternating path of H with respect to G between two exposed vertices
of H. An augmenting tail of H with respect to G is a n alternating path of H with respect
to G between a vertex vo which is exposed in G a n d a vertex vn+l which is covered in G ,
such t h a t un+l is covered by t h e G-triangle
~ ~ + l t ' ~ + 2 ~ Finally,
~ + 3 .
a n augmenting kite of
H with respect to G is a n even length alternating p a t h of H with respect t o G between
an uncovered vertex uo a n d some vertex vn covered by t h e G-edge v,,lvn,
a IC3
VnUn+l
vn+k+l, and a n alternating p a t h v,+l,
. . ., v,+k+l- A
together with
degenemte kite i s a kite
with n = 0 a n d k = 1. T h a t is, a triangle with o n e edge covered by the packing. T h e s e
augmenting configurations a r e shown in Figure 2.4. Note t h a t circled vertices denote vertices
which a r e exposed with respect t o t h e packing.
Augmentation of these configurations is d o n e in t h e following manner. For a n augmenting p a t h , we use a switching technique in which all G-edges a r e removed from G and n o n
G-edges a r e m a d e G-edges. T h i s switching technique is also used to augment t h e tail along
the p a t h t.0,.
.., v,+l.
In addition, in t h e c a s e o f t h e tail, edges vn+lvn+2 a n d un+lvn+3
become non-packing edges, b u t edge un+zvn+s remains a packing edge. T h e kite is augmented by switching along t h e p a t h s
VO,
.. ., vn, a n d
v,+l,.
..,Un+k+l, and by a d d i n g t h e
C'H.4PTER 2. SURVEY OF THE G E N E R A L C;RAPH PACIilNC: PROBLEM
1i3 v,vn+l
v,+k+l
to G.
Sow we present t h e !emma.
Lemma 2.1.1. r29] A {1<*, I<3)-pcking C of a yrnph H is of rnazimum size i j and only if
it admits no augmenting path, tail or kite.
Proof: Clearly, if G is of maximum size then there can be n o augmenting
configurations.
Suppose H is a smaliest graph which admits a packing G which has no augmenting configurations a n d another packing G' of II which covers more vertices than
G. Then there must be a vertex v which is covered by G' but not by G. We will
consider two cases:
1. T h e vertex v is covered by a G'-edge vw .
Since G has no augmenting configuration, v cannot be on a n edge with a
vertex which is exposed in G , and so w must be covered by
G.Since there
a r e no augmenting configurations, w cannot be covered by a G-triangle and
so must be covered by a G-edge wu.
E ( H) = E ( H ) \ {
edges incident with u or m), and the packings C and G' restricted to l?.
In fi, G covers two less vertices, namely m and u, and G' covers two less
vertices, namely v and w . Therefore, G' covers more vertices of K than
G and so by t h e choice of H, G must admit a n augmenting configuration
in H. If t h i s augmenting configuration begins at a vertex different from u
Consider t h e graph
fi with ~ ( f i=) V ( H )\ {v, m ) and
a n d does not involve u, then it would be a n augmenting configuration of
G in H a s well, but this contradicts the definition of G. If t h e augmenting
configuration begins a t a vertex other than u and ends a t u, then it is an
augmenting configuration ending at v in H. If t h e augmenting configuration
begins a t u, then in H we would have a n augmenting configuration of G
beginning at v with the edge vw and followed by the G-edge wu. -4gain
this contradicts the definition of G.
2. T h e vertex u is covered by a Gf-triangle vwt. Since G has n o augmenting
path o r tail, we cannot have any of the graphs of Figure 2.5.
C H A P T E R 2. SURVEY OF T H E GENER-4L C;R=ZPH PACKIiVC PROBLEM
Figure '2.5: Vertex
t. covered
Figure 2.6: Vertices w a n d
2
by a Gt-triangle
covered by G'-edges
Therefore, both w a n d z must be covered by G-edges. However, u: and z
cannot be incident with t h e s a m e G-edge since G has no degenerate kites.
Thus, there a r e two distinct G-edges as in Figure 2.6.
Consider the graph
fi with ~ ( f ? )= V(H)\{o,to, z } , a n d E ( H ) = E ( H ) \ {
edges incident with v , w , or z), and the packings
G a n d G' restricted to
a. In fi, G covers four less vertices t h a n in
H and G' covers three less
t h a n G. So, by t h e choice
vertices. Therefore, G' covers more vertices of
of H , G must have a n augmenting configuration in H. If t h e augmenting
configuration is a p a t h between z and y, then in H we would have t h e
augmenting configuration of G shown in Figure 2.7, which is a n augmenting
kite in H. This is a contradiction with t h e definition of G.
If t h e augmenting configuration begins at x (or y), then i t is a n augmenting
configuration of G beginning a t u in
N ,which
again contradicts t h e choice
of G. If t h e configuration begins at some point o t h e r t h a n x o r y, then it
is a n augmenting configuration of G in H as well, a n o t h e r contradiction.
Therefore, the packing G must be of maximum size. C3
T h e augmenting configurations of Lemma 2.2.1 c a n be used in a n algorithm t o find
C'Hk\PTER 2. SURVE k' OF THE C;ENERA L GR.4PH P..1CIiI!VG PROBLEl\d
17
Figure 2.7: Augmenting p a t h between x a n d y
a masimum size packing in polynomial time. Before we give t h e algorithm, we neecl t h e
foIlowing proposition which uses some new terminology. A set of vertex disjoint triangles of
H will be denoted T.
Suppose t h a t a given packing G is not of maximum size. T h e n it follows from Lemma 2.1.1
t h a t G a d m i t s a n augmenting path, tail o r kite a n d therefore there is a larger packing C'
whose s e t of triangles differs by a t most one triangle from t h e set of triangles of G. This is
written formally in t h e following theorem.
Proposition 2.1.1. ['29] Let G be a {Kz,
K3)-packing o/ H, 7 the set of C-triangles. Then
G is a {Kz,
Ks)-packing of maximum size if and only if no {K2,
Ks)-packing G' of H cocers
more vertices than G where the set of G'-triangles:
I . is equal to T :
2. is equal to T \ { u v ~ wfor
) some lnbngle { u v w ) E T :
3. is equal to T U{ u v w ) for some triangle (uvw)
T.
Proof: If G is of m a ~ i r n u msize, then clearly there is n o packing G' covering
more vertices t h a n G.
Suppose t h a t G is not of maximum size. T h e n by Lemma 2.1.1 there must be a n
augmenting configuration resulting in a larger packing G'. If t h e configuration
were a p a t h , then G' would have t h e s a m e number of triangles as G (case I ) .
If t h e configuration were a tail, then G' w o d d have one less triangle than G
(case 2). Finally, if t h e configuration were a kite, then G' would have one more
triangle t h a n G (case 3). 0
Proposition 2.1.1 a n d Lemma 2.1.1 lead to a polynomial-time algorithm for finding a
maximum {K2,
K3)-packing. In this algorithm, G denotes a
denotes the set of G-triangles.
{K2,
&)-packing of H , and 7
C'HAPTER 2. SURVEY OF THE GENERAL GRAPH P.4CIifiVG PR0BLEA.I
Algorithm 2.1.1. To find a mazimum
begin
G':= maximum matching of H
size
{ I C 2 , K 3 ) - p c k i n g of a gmph H .
I-:= 0
while (mazimum size of a {I<*, K3)-packing C' with set Tf o j GI-triangles, where T' =
7 \ { u v w ) for some { u v w ) E 7, or T r= T
G':= G'
T:=T
return G
U {uu.w} jor some { u v w ) #
7)> size
of G' do
end
In this algorithm, t h e s e t of triangles, T' of t h e packing G' is related to t h e set of triangles
'J of G as follows: 7' = 7 \ {uvw), where u v w E
7,
or T
= T u { u v w ) , where u v w @ 7.
T h e s e t T' a n d consequently t h e packing G', c a n be found in o n e of two ways. Firstly, we
systematically choose one triangle uvw from 'J a n d remove it from T t o o b t a i n T-. We
then run a maximum matching algorithm o n H \T-. T h e resulting matching together with
T- gives us a packing. If this packing covers more vertices t h a n G, then it is a packing G'
a n d T- = T'. We then update G a n d
7 a n d continue. If not, we choose a different triangle
of T t o remove a n d proceed as before. If n o larger packing G' is found, we move o n to t h e
triangles which a r e n o t in T. T h i s time, we systematically a d d a triangle uvw which is not
in 7 t o
7 t o o b t a i n T+.Then
as before, we r u n a maximum matching algorithm o n H \ T+
a n d t h e resulting matching along with 'T+ gives u s a packing. If this packing is larger t h a n
G , t h e n it is a packing G' a n d T+ = T'. W e t h e n update G a n d T a n d r u n t h e while loop
again. If no packing G' is found which i s larger t h a n G , then G is a maximum packing a n d
we a r e done. T h i s process is polynomial, since if n = V ( H ) ,t h e number of triangles in T
a t a n y time i s at m o s t order n, as n o vertices can b e repeated, t h e number of triangles n o t
in 7 at a n y time is at most order n3, a n d finding a maximum matching of t h e remaining
vertices c a n be found in polynomial t i m e [20].
Proposition 2.1.1, L e m m a 2.1.1, a n d t h e algorithm can be extended t o o t h e r cases including
t h e case of Theorem 2.1.4 below where G = {K2,
F l y . ..,Fk)a n d all t h e
able. A g r a p h H is hypomatchable
Fia r e hypomatchif H has n o perfect matching, b u t H \ v d o e s have a
CH.4 PTER 2. SURVEY OF THE GENERAL C;RA PN P.4CKIiVG PROBLEM
Figure 2.8: T h e kite for G = {Ir;,C;)
perfect matching for all v E V ( H ) .
Theorem 2.1.4. [29] Let G = ( Kz,F , , . ..,F k ) and assume that each Fi is hypomatchable.
Then there exists a polynomial-time algorithm to find a maximum site G-packing of arbitrary
input g m p h H .
This theorem will not be proven here. The proof given in [29] uses a n extension of
Lemma 2.1.1. In order to extend Lemma 2.1.1. we need t o extend t h e augmenting configurations t o the case G = (Icz, F I T ...,Fk).Let G a n d G" be 6-packings and let an alternating
path refer t o a G, G'-alternating path in H whose edges are alternately G-edges and C"-edges.
T h e augmenting configurations are then as follows.
1. An augmenting path is a n atternating path between two vertices exposed in G.
2. An augmenting tail is a n alternating p a t h between a n exposed vertex of G and a
vertex covered in G by some G - Fi (not by a G-edge).
3. An augmenting kite is a n even length alternating path between a n exposed vertex u of
Fisuch t h a t t h e Fiand the alternating path intersect only
Also, there exist $(IV ( F i I) - 1) vertex-disjoint alternating paths between pairs
G and a vertex
at v.
of vertices of Fi
v of some
- u such t h a t each path begins and ends with a G-edge. An example
of such a kite is given in Figure 2.8.
In this Figure, the
f ((v(F;)
( - 1) vertex-disjoint
alternating p a t h s are the paths 1 , 3 ,
and 2,6,5,4.
T h e following proposition is analogous to Lemma 2.1.1.
C'H-4P T E R 2. SURVEY OF T H E GENERAL GRA PH P..\CI\'ILVG' PROBL EAf
20
Proposition 2.1.2. 1291 Lei G satisfy the clssumptions of Theomm 2.1.4. The G-packing
G of a graph H is of mazimum size if and only if it has no augmenting p l h . tail? o r kite
in [ I .
T h e r e is a corollary t o Theorem 2.1 .-l which considers a possibly infinite family of cliques.
C {K1,
K 2 , K 3 , . . .) (G not necessan'ly finite) and
E G, then there is a polynomial-time algorithm for the maximum site G -
Corollary 2.1.2. [27. 29, 17, 361 If
Ii-, E 6 or
packing problem.
P m f : Clearly K r E G is trivial and
G
=
1'-2
is solvable by a polynomial
m a s i m u m matching algorithm. T h e n consider t h e case where K 2 is in
G and
all o t h e r cliques in G a r e even. All elements of
so a
G have perfect matchings a n d
maximum size G-packing is found by searching for a maximum size (K2)-
packing.
If K 2 is in
6 a n d Kt, t > 1, is t h e smallest odd clique in G,then a n y G-packing
h a s a ( K 2 , Kt)-factor since o d d cliques c a n b e covered with a copy of
and
s o m e number of copies of K 2 , a n d any even clique can be covered with copies of
IC2. Therefore: i t is sufficient to find a maximum size { K 2 ,Kt)-packing which
by Theorem 2.1.4 c a n be d o n e in polynomial time.
0
Families of Complete Bipartite Graphs
2.2
Recall t h a t a complete bipartite g r a p h having a partition of size o n e a n d t h e o t h e r of size
k is called a star a n d is denoted
Sk.
T h e sets {Sly
S2,.
. .,S k ) a n d { S r , Sz, .. .) a r e called
sequential star sets.
.
T h e search for a polynomial algorithm t o find a maximum {S,,. . , Sk)-packing s t e m m e d
from a n observation of Las Vergnas, given below, concerning (1, k)-factors. A ( 1 , k)-factor
is a spanning subgraph with all degrees between 1 a n d k.
Proposition 2.2.1. [47] A graph h a s a {S1,. .. , S k ) - f a c t o r if and only if it has a ( 1 , k)factor.
Proof: Given a graph H which h a s no isolated vertices, consider t h e following
O(I El) algorithm.
C'H-4PTER 2. SURVEY OF T H E GENERAL GR4 PH PACf\'INC PROBLEhl
21
Algorithm 2.2.1. To j n d a {S1.. ...Sk)-/actor o r {Sl, S z , .. .)-factor of H .
I . F o r each edge trv f H , delete u v if both u and v h a w degree greater lhan
one.
2. Update the degrees of u a n d v.
If all degrees of t h e vertices of H a r e between 1 and k, t h e n this algorithm
. . ..Sk)-factor.
will find an {Sl?
T h u s a (1. k)-factor found by a n y algorithm
for finding (1, k)-factors ['LZ, 441,can be modified t o give a n (SI,.. .,Sk)-factor.
Since t h e ( 1 , k)-factor problem is polynomial, t h e { S l , .
. .,Sk)-factor problem is
also polynomial.
Note t h a t if the algorithm is applied t o a n arbitrary graph H , t h e n it will find
a maximum sequential s t a r packing, t h a t is a n {Sly
Sz, . ..)-packing. Therefore.
t h e {Sly
Sz, . ..)-packing problem is polynomial.
I t is proven in [28, 301 t h a t for a n y o t h e r families
G
of complete bipartite graphs, t h e
G-packing problem is N P - c o m p l e t e o r harder. We will not prove this here.
Proposition 2.2.2. [28, 301 A graph II has an {Sl!S 2 , . . .)-factor if a n d only if it has n o
isolated vertices.
Proof: This foIlows from Proposition 2.2.1. Since each vertex m u s t have degree
at least 1, no vertex c a n b e isolated. O
T h e augmenting configurations referred to in Theorem 2.2.1 beiow a r e shown in Figure 2.9. T h e number o r letter beside each vertex denotes t h e degree of t h a t vertex in t h e
packing. Brackets with a n asterisk a r o u n d a p a r t of t h e configuration denote t h a t t h a t part
of t h e configuration may be repeated q times, q
> 0.
As before, bold edges denote edges
covered by t h e packing a n d circled vertices denote vertices uncovered in t h e packing.
When we augment these configurations, we obtain the configurations of Figure 2.10.
Theorem 2.2.1. [2R, 30) Let k
2 2.
The size of a mazimum {Sly..
.,Sk}-packing o / H is
equal t o the minimum, over all T C_ V ( H ) , of n
denotes the number of isolated vertices of H
\ T.
+ k - IT1 - ir
where n = Ik'(H)[ a n d i~
CHAPTER 2. SCIRVE)' OF THE GENERAL GRAPH P-~CKIIVG
PROBLEM
(ii)
22
(iii)
Figure 2.9: Augmenting s t a r configurations
(9
(i i)
(iii)
Figure 2.10: Augmented s t a r configurations
Proof: If T is a subset of V ( H ) , t h e n t h e largest number of vertices in H
\T
which a r e not exposed in H occurs if each vertex in T is t h e c e n t r e vertex
of an Sk a n d all t h e o t h e r vertices of t h e Sk a r e in H
\ T.
T h i s n u m b e r is
k ITI. Therefore, t h e number of exposed vertices in t h e packing G o f H is
at least t h e number of vertices which a r e exposed in H a n d t h i s n u m b e r is
i~ - k - ITl. Therefore, t h e n u m b e r of covered vertices in t h e packing G is at
most n - (iT - k ITI) = n + k . IT1 - i~ for any 2' S V ( H ) .Now, to prove this
equality we define a packing G a n d a s e t T C V ( H ) such t h a t G covers exactly
n + k - IT1 - i~ vertices of H .
Let G b e a n y {Sl, .. ., Sk}-packing of H which a d m i t s no a u g m e n t i n g configuration. Let U d e n o t e t h e s e t of exposed vertices in G. Let t h e sets T a n d PV b e
defined as follows:
If degcv = k a n d
u is adjacent to a vertex of
U U W in H , then v
f T.
CH.4PTER 2. SURVEY OF T H E GENERAL GRAPH P.AC'f\'lrVG PROBLEA9l
If w is adjacent t o a vertex of T in G, then w E 1.2'.
Since G a d m i t s n o a u g m e n t i n g configuration, each neighbour of a v e r t e s u E U
must have degree k with respect t o G' a n d so b e a n element o n T . .;\lso. a n y
bV must have degree 1 in G a n d be reached from s o m e u by a p a t h o n which
t h e degrees of t h e vertices a l t e r n a t e between k a n d 1 with t h e vertes adjacent t o
w E
k. Since t h e vertes
preceding w has degree k a n d is adjacent t o w € W , i t must be in T a n d so each
neighbour of a w f W is a n element of T. Consequently, all vertices of Cr u W
must b e isolated in H \ T, a n d s o i~ 2 IU( + IW(. T h i s means t h a t G covers
n - lUl > n + l W I - i~ = n + k - IT1 -iT vertices. O
u having degree k and t h e v e r t e s preceding w having degree
Corollary 2.2.1. [28, 301 An {S1, . ..,Sk)-packing G of H is maximum if and only if it
admils no augmenting configurntion.
P m f : Clearly, if G a d m i t s a n augmenting configuration then G is not maximum. If G does n o t a d m i t a n augmenting configuration, then by the proof of
Theorem 2.2.1, t h e r e exists a set T C V ( H ) such t h a t G covers n
+ k - IT1 - iT
vertices of H a n d s o is a m a x i m u m packing. 0
Another corollary t o Theorem 2.2.1, given beIow, w a s proven independently by Amahashi
a n d K a n o f6] a n d by Las Vergnas [47]. I t c a n also b e found in [3, 91.
Corollary 2.2.2. ['La, 301 Let k
2 2. A
graph H admils an {Sly
.. .,Sk)-factor ijand only
if it does not have a set T of vertices whose deletion results in more than k [TI isolated
ue rt ices.
P m f : ['28, 301 B y Theorem 2.2.1, t h e maximum size of a packing of H is t h e
minimum of n + k 17'1 - iT over all T. If there is a set T such t h a t iT > k - [TI,
then n + k - IT1 - i~ < n. Therefore, t h e maximum size of a packing will be n if
a n d only if it 5 A:. 17'1 for each T & V ( H ) . CI
2.3
Families of Paths
Having considered families of c o m p l e t e g r a p h s a n d of corn plete bipartite graphs, we now
shift o u r attention to families of paths. Recall t h a t in t h i s thesis,
with k edges.
Pk represents t h e p a t h
CWAPTER 2. SURVEY' OF THE GEXERAL GRAPH P.4C,'i<IArG PROBLEM
T h e maximum G-packing problem where
G is t h e p a t h o n o n e edge, P I , is equivalent t o
the niasirnum matching problem which is well known t o be polynomial.
The Family {Pz}
I t follows from Theorem 2.1.1 t h a t t h e {P2}-packing problem is N P - c o m p l e t e . We now give
a proof of this using t h e local replacement technique.
Theorem 2.3.1. The (P2)-factor problem is JVP-complete.
Proof: This problem is in AfP since if we a r e given a potential {P2)-factor of
a graph H as a certificate, we c a n verify t h a t all vertices of H a r e covered by
exactly one P2 in polynomial time. We will reduce t h e N P - c o m p l e t e problem
3DM t o t h e {P2}-factor problem to establish t h e NP-completeness. Consider
t h e gadget of Figure 2.1 1.
Figure 2.11: G a d g e t for t h e {Pz)-factor problem
7- = { ( t l , t z lt 3 ) l t l E W , t 2 E
X,t3 E Y } , s o m e s u b s e t of W x X x Y where W ,X,Y = (1,2,. ..,g ) . W e will
construct a g r a p h G as follows: Let W U X u Y be t h e vertices of a g r a p h . For
Construction: Recall t h a t 3DM considers a s e t
each triple ( t l ,t 2 ,t 3 ) of T ,identify t h e connector vertices, a, 6, a n d c, of a copy
of t h e gadget with t h e corresponding vertices t l , t 2 a n d t3 of t h e graph. T h e
edges of these copies o f t h e gadget a r e in t h e edge set of t h e g r a p h a n d t h e
interior vertices a r e in t h e vertex set. Call t h e resalting g r a p h G.
Claim 2.3.1. The set T has a 3-dimensional matching if and only if G has a
(P2)-Jacior.
C,'HAPTER 2. SURVE)' OF THE GENERAL GRAPM PACKllVC PROBLELLI
Proof: Suppose 7- h a s a 8-dimensional matching. T h e n for chosen
triples ( t tz, t 3 ) , Figure 2.12 shows t h e { P2)-factor of t h e copy of t h e
gadget having connector vertices t l = a. t2 - 6 a n d tJ = C.
F i g u r e 2.12: A { Pz)-factor containing t h e connector vertices
For unchosen triples, t h a t is when 11, t 2 , a n d t S a r e each covered by
s o m e o t h e r triple, t h e copy of t h e gadget having connector vertices
t l = a , t 2 = b a n d t S = C, h a s a {P2)-factor as shown in Figure 2.13.
Figure 2.13: A {P2)-factor excluding t h e connector vertices
T h u s , t h e r e is always a {P2)-factor of G when there is a 3-dimensional
matching of T .
Suppose G h a s a {P2)-factor. Consider a copy of t h e gadget. If vertex
a is covered b y t h e P221x20, t h e n a' c a n be covered by choosing afb'yz
If we choose a'b'y2 then there is n o way t o cover yl. T h u s , we
2
must choose a'b'c' a n d so b will b e covered by y1y2b and c by 2 1 ~ as
in Figure 2.12. In this case, t h e triple (a,6, c ) is in t h e 3-dimensional
o r a'b'd.
matching.
If instead a is covered in s o m e o t h e r way, t h e n
by t h e
XI
must be covered
P2 x1x2a'. Thus, 6' c a n b e covered by yly2bf, by btc'z2, o r by
~
2.5
CH-4PTER 2. SURVEY OF THE GENERAL GRAPH PACKlNC' PROBLEM
26
1 ~ ~ 6 'Ind .t h e last case, we a r e unable t o cover vertes yl. Choosing
6 ' 8 ~ wotild
~
leave t h e vertex
uncovered in t h e packing a n d so we
must choose ~ ~ 3 ~ 6 Therefore,
'b a n d c will be covered by Pzs other
t h a n y1y26 a n d 21z2c as in Figure 2.13. In this case, t h e triple (a,b, c )
is n o t chosen as a triple of t h e 3-dimensional matching.
T h e resulting set of chosen triples 7- = {(a,b, c ) ) is 3-dimensional
matching. CI
Since t h e iVP-complete problem 3DM reduces t o t h e {P2)-factor problem, the
{ Pz)-factor problem is also NP-complete. 0
An interesting class of input graphs for which t h e {P2)-packing problem has been studied
is t h e class of squam graphs. Let i E ( 1 , . .., n} a n d j E (1,.
. ., m).
Consider the grid g r a p h
V = (1, . . .,n) x (1, . ..,m) with n m vertices such t h a t for i, k E (1, . . .,n ) a n d
j, 1 E ( I , . . .,m ) , vertex (i,j) is adjacent t o vertex (k, 1 ) if and only if i = A: and Ij - 11 = 1
or li - kl = 1 a n d j = 1. A square g m p h is a subgraph C of t h e grid graph with the property
defined by
t h a t G is connected and each edge of G is contained in some C; of G. An example of a
square graph with a (&)-packing is given in Figure 2.14.
Figure 2.14: A {P2)-Packing of a Square G r a p h
Some results concerning square graphs and their {P2)-factors a r e given in the next
theorem.
Theorem 2.3.2. [5] Let G be a square gmph on p vertices.
(I) If p
(11)
0 (mod 3 ) , G has a
{P2)
-factor.
If p r 1 ( m o d 3) and v is an arbitrary vertex ojdegree 2 in G,G\ v has a {Pz)-factor.
T h e Family { P I ,P2)
Results o n the packing problem for t h e fami[?; { P I ,P2) can be found in [3, 1. 2. 51.
.An interesting result o n ( P , , Pz)-factors. Corollary 2.3.1, concerns triangle graphs. Consider a triangulated grid. -4 connected subgraph of this grid in which every vertex is contained in a IT3, is called a t rimgle graph.
We have t h e foIlowing corollary concerning { P I ,Pz)-factors of triangIe graphs. Such a
factor is shown in Figure 2.15. This corollary was proven by -4kiyama and Kano.
Figure 2.15: A { P t,P2)-Factor of a Triangle G r a p h
Corollary 2.3.1. [.5] Every triangle graph T has a { P I .P2)-factor.
Proof: Let S be a vertex subset of T. A vertes x of T is incident with at most
six vertices. Thus, t h e maximum number of vertices incident with z which can
be isotated in T \ S is four, as shown in Figure 2.16.
Figure 2.16: -Maximum isolation of vertices incident with x
Also, each isolated vertex of T \ S must be incident with at Ieast two vertices of
S. Therefore,
CHAPTER 2. SURVEY-OF THE G E N E R A L GRAPH P.4CKlNC: PROBLEM
28
where. i(T \ S ) is the number of isolated vertices in T \ S.
T h u s , by Corollary 2.2.2, with k = 2. a triangIe g r a p h must admit a { P , , P2)factor.
2.4
Families of Cycles
T h e two-factor problem, which is t h e 6-factor problem where
nomial
p24.
G = {C3,C4,..
.)?
is poly-
In addition t o this, t h e g r a p h packing problem is polynomial for t h e family
= {C4,C;,. ..), see [31] a n d for t h e family G = { G , C 5 , C 6.,-. ) [7]. T h e g r a p h packing problem for t h e family G = {G,G,..- ) is expected t o b e polynomial, see [31], b u t
this problem remains open. For all o t h e r finite families of cycles, t h e packing problem is
N P - c o m p l e t e r29. 311. T h e JVP-completeness proofs for t h e packing problem with families
= {C3tC4,C6,C;T...),
a n d = {C4,C6,C?, ...) c a n b e found in [18],
a n d with t h e family G = {C3,C5,C;, . . .) c a n b e found in [ G I .
G = { C 6 , C 7 , . . .),
Two-Factors
A (C3,
Cl, .. .)-factor, of a graph H is a s u b g r a p h F of H for which V ( F ) = V ( H ) a n d
every vertex h a s degree two. Thus, a two-factor of H consists of disjoint cycles t h a t cover
1'- (H).
Theorem 2.4.1. [46] A gmph G = (V, E) a d m i t s a {C3,
C4,. ..)-factor if and only if
where IC is the s e t of vertices ofdegree at most 1 in G\A,
and
x
is the number of cornponenls
of G \ A which contain an odd number of vertices from I<.
2.5
Families of Mixed Graphs
T h e packing problem where t h e farniIy
G
consists of I<;! along with some o t h e r g r a p h was
studied by Cornut5jols, Hartvigsen a n d Pulleyblank [IT], by Hell a n d Kirkpatrick r28, 36, 29,
301 a n d by Loebl a n d Poljak [39, 401. Loebl a n d Poljak have completely characterized t h e
complexity of deciding whether a perfect packing exists for such families.
CH.4 PTER 2. SURVEY OF THE GENER4 L G R A P H PACIiliVG PROBLEM
29
Some early work o n t h e { K 2 ,F)-packing problem studied t h e problem for F = P2
[ 3 ? 28. 471 a n d for F hypomatchable [IT. 16, 291. In b o t h cases t h e packing problem is poly-
nomial. These results gave hope for other polynoniial results. T h e main polynomial result
of this section, Theorem 2.5.1, concerns t h e class of g r a p h s F which a r e perfectly matchable,
hypomatchable o r propellers. A propeller is a g r a p h obtained from a hypomatchable graph
I+- by adding two new vertices, c? t h e centre. a n d r , t h e root, the edge cr a n d some edge(s)
from c to a nonempty subset of IY.
Theorem 2.5.1. [41, 401 The {K2,
F)-packing ~roblernis polynornially solvable /or all
gmphs F which are either perfectly matchable, hypomatchable, o r a propeller. The pmbfern is NP-complete for all other gmphs F .
I t is clear t h a t if all g r a p h s F a r e perfectly matchable, then t h e problem is equivalent t o
t h e maximum matching problem. The theorem was proven in [li]for g r a p h s F which are
hypomatchable and in [41] for graphs F which a r e propellers. These two proofs will not be
presented here.
2.6
Applications
-4s we have mentioned in C h a p t e r 1, matching theory has many real-world applications.
This theory has been used in scheduling, assignment, transportation, network flow. shortest
p a t h , travelling salesman and Chinese postman problems.
A specific application for the graph packing problem is the a r e a of examination schecluling. In [27], Hell a n d Kirkpatrick consider such a problem. T h e problem of minimizing the
number of exam periods a n d avoiding conflicts such as students taking two e x a m s a t a time,
is a well-studied problem a n d is NP-complete. T h e problem studied in [27]is t h a t of g r o u p
ing exam periods into sets of k periods. Consider a set of courses which have already been
separated into examination periods. T h e k periods represent k examination days, a n d the
exams in a given period a r e organized within t h a t day's schedule. T h e goal is t o minimize
t h e second order conflicts, such as a student being required t o sit two e x a m s o n t h e same
d a y o r with only a given amount of time between exams. This problem c a n be formulated
in a g r a p h theoretic way.
C H A P T E R 2. SURVEY OF T H E GEXER.4L GRAPH P.4CIIiING PROBLEM
Inslance: -4 graph G. a positive integer n a n d w a weight function o n t h e graph
such t h a t
LJ
: E(Ii,)
K,
+ LO, 00).
Question: Are there d disjoint s u b g r a p h s C1,G2,. ...C'd of
each
30
G iis isomorphic
such t h a t
to G'.
each vertex of Kn is in s o m e
Gi,and
d
2 Cee.q~.)
w ( e ) is minimized?
i= 1
In this definition of t h e problem. t h e graph G, with k vertices, corresponds t o the
definition of t h e second order conflicts. T h e g r a p h I<, represent t h e n examination periods
and we c a n assume t h a t n = kd. T h e weight function, w ( e ) , where e = ( u , v ) , corresponds t o
t h e number of students w h o need t o write a n e x a m in both period u a n d period u . Consider
t h e example used in [27]. If G =
K3,then k = 3 and t h e solution G I , . . ., G d ,corresponds
t o a partition of t h e examination periods into d d a y s with 3 periods per day. T h e number
of occurences of a student being required t o write two consecutive e x a m s o n t h e s a m e d a y
is minimized.
A s we can see from t h e question stated for this problem, we a r e looking for a G-factor
of I<,. Notice t h a t in t h e formulation discussed above, it is assumed t h a t t h e scheduler has
eliminated first order conflicts.
Chapter 3
Directed Graph Packings
S t u d y o f t h e graph packing problem for undirected graphs resulted in many polynomial
algorithms as well a s some JVP-completeness results. We now move o u r focus to t h e graph
packing problem for directed graphs.
.4fter introducing t h e definitions a n d notation t o be used in t h i s chapter, we will look
a t t h e directed graph packing problem for directed paths, cycles a n d s t a r s .
3.1
Definitions and Notation
Recall from C h a p t e r 1 t h a t a directed graph o r digraph H = ( V ( H ) ,A ( N ) ) is a graph with
t h e s e t V ( H ) of vertices a n d t h e s e t A ( H ) of arcs. Recall t h a t a n a r c a6 is oriented from
u t o 6. If we a r e n o t interested in t h e direction, o r d o not know the direction on a given
arc, we will refer to i t simply as a n edge. A directed G-packing of a digraph H is a s e t of
pairwise vertex-disjoint subgraphs of H each of which is isomorphic to s o m e digraph in t h e
family G. A vertex of H is covered by t h e directed G-packing if i t belongs t o a subgraph of
t h e G-packing. If t h e directed G-packing h a s size IV(H)Iythen it is a directed G-factor.
T h e directed G-factor problem c a n be s t a t e d a s follows:
Given t h e family of directed g r a p h s 6,
Directed G-factor:
Instance:
A digraph H .
Question:
Does H a d m i t a directed G-factor, t h a t is, is there a packing
of H using digraphs from t h e family G such t h a t t h e size of
t h e packing is IV(H)I?
C'H.4 P T E R 3. DIRECTED CRAPH PACl\'lhrC;S
:3 2
For t h e remainder of this chapter. the terms G-packing a n d 6-factor will refer to directed
G-packing and directed G-factor unless otherwise stated.
-AS we have mentioned, we will s t u d y this problem for directed paths. cycles and stars.
.-\ direcled lmth o n k edges, denoted
fi. is a g r a p h
{ ~ ; v ; li
+ ~E {I, .. .,L)}. A directed cycle o n
with vertex s e t {vl, . . ., a k + l ] a n d arcs
k edges, denoted
{ v I, . . ., uk} a n d arcs {vl up,u2U3, - - ., uk-1 vk, ukvI}.
denoted
s;,
G , is a graph with
vertices
Finally, a directed star on A- edges,
is a complete bipartite graph with partitions {u) a n d { v l . . .., v k } and arcs
{uv,Ii € { I , . ..,k)}.
T h e underlying g r a p h of a directed graph H is t h e g r a p h H' o n t h e s a m e vertex set as
N. T h e edge
uv is in
E(H') if u v E A ( H ) o r v u E A ( H ) o r both.
Packing with Directed Paths
3.2
3.2.1
One-Element Families
The Family 6 = {PI}
Theorem 3.2.1. The maximum {fi}-packing problem is i n P.
Proof: For a digraph H, consider t h e underlying undirected g r a p h H'. T h e n
the { a } - p a c k i n g problem for H becomes t h e maximum matching problem for
H', which is known to be polynomial. Therefore, if we find a maximum matching
of t h e g r a p h H', t h e n by choosing t h e arcs of H corresponding t o t h e matched
edges of H', we o b t a i n a maximum {fi}-packing of H in polynomial time. O
The Family
C = {fi}
Theorem 3.2.2. The
{e}-factor problem is NP-complete.
P m f : T h e proof of this theorem is analogous t o t h e proof of Theorem 2.3.1. If
we have a {p2}-factor as a certificate, we c a n verify i t s correctness in polynomial
time. Therefore, t h e {&}-factor problem is in
t o the
(6)-factor
N P . W e will now reduce DDM
problem. Consider t h e gadget of Figure 3.1.
Recall t h a t an instance of 3DM is a set of triples T = {(t t 2 ,t3)lt E W ,t Z E
X,ts E Y ) which is s o m e subset of W x X x Y (W,X , Y = { 1 , 2 , . ..,n}). We
CH.APTER 3. DI R EC'TED GR.4 PH P..IC.'f<flVCS
Figure 3.1: G a d g e t for t h e { e l - f a c t o r problem
will construct a graph G such t h a t 7 h a s a 3-dimensional matching if a n d only
if G a d m i t s a {GI-factor.
Construction: Let t h e vertex s e t of a g r a p h consist of o n e vertex for each element
of
M'
U ,Y
uY.
For each triple of T , a d d a copy of t h e gadget to t h e graph by
identifying connector vertex a with vertex t l , connector vertex b with t2, and
connector vertes c with t 3 . T h e interior vertices a n d the edges of t h e copy of
t h e gadget become vertices a n d edges of t h e graph. Call this graph G .
Claim 3.2.1. The set 7- has a 3-dimensional matching if and only if G has a
{el-factor-
P m f : Suppose ';r h a s a 3-dimensional matching. T h e n for triples
( t Lt,2 ,t 3 ) in t h e matching we have t h e {p2)-factor of t h e copy of t h e
gadget, having connector vertices t l = a,ta = b, and t3 = c, shown in
Figure 3.2.
Figure 3.2: A {p2)-factor containing the connector vertices
C'N.4 PTER 3. DIRECTED GRAPH PACIiINGS
For triples not in the matching, t h a t is when t I , t 2 , and t 3 are each
covered by some other triple, the correspondirig copy of the gadget in
G has a { e l - f a c t o r as in Figure 3.3.
Figure 3.3: A ( 6 ) - f a c t o r excluding t h e connector vertices
Thus, there is always a
{fi }-factor of G when there is a Mirnensional
matching of 7.
Suppose G h a s a (6)-factor.
Consider a copy of the gadget. T h e
vertes x 1 can be covered by choosing t h e directed path xlx2a o r the
directed path xlx2at. If it is covered by ~ 1 x then
2 ~in ~order t o cover
a', atbtc' m u s t b e in the packing. This in t u r n forces yly2b and zlzzc
to be in t h e packing. Thus, if one of the directed paths incident with
a connector vertex is in the packing, then the connector vertices of
the gadget a r e covered a s in Figure 3.2. In this case, (a,b, c ) is chosen
as one of t h e triples in the 3-dimensional matching. If
XI
had been
covered by t h e directed path x1z2at, then t h e only way t o cover b'
would have been to choose the directed path y ly2bt and this would
force t l z 2 c ' to b e in the packing as well. T h u s if t h e directed path of a
gadget incident with a connector vertex is not covered by the packing,
then none of t h e connector vertices of t h e gadget will be covered by
the packing. In this case, the triple (a,6, c) is not in the 3-dimensional
matching.
T h e resulting set of chosen triples is a 3-dimensional matching.
Thus, the ( 6 ) - f a c t o r problem is NP-complete.
a
C'H.4 PTER 3, DIRECTED G R4 PN P.~CI<IIVGS
3.5
T h e gadget used for t h e ( 6 ) - f a c t o r problem can b e used for o t h e r directed paths of
length two by altering t h e a r c directions in the paths x1x2n. xlxza', a1 G1 c1 , yly2b. 91y2b'.
z1z2c,
and z I z 2 c 1accordingly.
completeness of the $;-packing
O n e such nlodification of the gadget is used in t h e N P problem on page 64.
The main result of this chapter concerns t h e graph packing problem with t h e family
{fi: f3}
a n d will be given in Section 3.3.
T h e graph factor problem with the family
{PI,&}
{el-
is actually t h e s a m e as the
factor problem since a n y copy of f?! can be perfectly packed with t w o copies of t h e graph
.
all
Therefore, t h e
{pl,4 ) - f a c t o r
problem is polynomial. A similar argument shows t h a t
{ply
6 )-factor
problems where j is odd a r e polynomial.
We will now consider t h e { f i ,El}-factor problem. O u r approach is based upon work by
Loebl a n d Poljak in [dl]. We will consider the
{fi,&}-factor
problem with a restriction on
t h e vertex s e t of the i n p u t graph. If this restricted problem is JVP-complete, and we will
show t h a t it is, then t h e
{PI,q ) - f a c t o r
problem is also JVP-complete.
In the restricted { P I , fi)-factor problem stated below, the vertex set V ( G )of input
graph G is made u p of t h e disjoint union of a set D a n d a s e t .4 of vertices. T h e set D is
independent, which means t h a t for all u, u E Dl uu
restricted
{ f i ,fi)-factor:
4 E(G).
Inslanee:
Digraph G with V(G) = D(G)u A ( G ) ,
such t h a t D is independent and 2101 =
Question:
Does G a d m i t a
314
{PI,fi}-factor?
Proposition 3.2.1. A digraph G with V ( G )= D ( G ) U .4(G)! such that D is independent
and 21DI = 31AI has a
each of the k
{fi,fi}-/actor
i j and only if G admits a { e l - j a c t o r F such (hat for
fis 01F, vertices U I , u3, and US a m vertices of D
and vertices
u2
and
u4
vertices o j A.
Proof: Consider a digraph G with V ( G )= D ( G ) u .4(G), such t h a t D is independent and 2101 = 31AI. If there is a {fi}-factor F of G, with t h e requirement
stated in t h e proposition, t h e n there is a {fi,fi)-factor a n d there a r e 3k vertices
in D and 2k in A.
are
CHA PTER .3. DIRECTED GR-4PH PACI\'INC;S
Suppose t h a t we have a
{fi. fi}-factor.
36
We will prove by counting that t h e
vertices in D covered by t h e factor. a r e covered by f i ' s meeting t h e requirement.
Suppose t h a t t h e factor uses i 6 ' s such t h a t vertices
of D a n d vertices
of
D ;n
"2,
and
"4
vl
, Q, and
us cover vertices
cover vertices of A: j fi's which cover two vertices
6's which cover one vertex o f D; p fi's
which cover n o vertices of
D:1
6's such t h a t o n e vertex covers a vertex of .4 a n d t h e o t h e r a vertex of D; a n d rn
P1'ssuch t h a t b o t h vertices cover vertices of A. Since 2101 = 4-41, we have t h e
=
2(Ri+2j+n+l)
6i+4j+'Ln+21
following:
=
3(2i+3j+4n+5y+1+2m)
6 i + 9 j + 1 2 n + I.5p+31+6nz
0=.5j+10n+1.5p+1+6m
Since j, n , p, 1 a n d m a r e non negative integers. t h e only solution is t h a t j =
4
n = p = 1 = m = 0 a n d so we only use P4's obeying t h e requirement in t h e
factor a n d so we have a ( 6 ) - f a c t o r of G. 0
Consider t h e colouring of a
6 shown in Figure 3.4.
Figure 3.4: Colouring o f a
fi used in C O L { ~ ~ }
We define C O L { ~ ~as} follows:
COL-{fi}:Instance:
A directed g r a p h G a n d a colouring 4~ : V ( G )+
(0, I} of t h e vertices of G.
Question:
Does G admit a {GI-factor which is faithful to t h e
colouring?
Theorem 3.2.3. The
P-f:
C O L - { f i } -/aclor problem
is AfP-complele.
We will reduce 3DM to t h e COL-{el-factor
problem. Consider t h e
g a d g e t of Figure 3.5 with connector vertex b a n d connector edges a a n d c.
Consider a n instance of 3DM for which t h e s e t s W and Y consist of q a r c s o n
t w o vertices each, a n d t h e s e t X consists of 9 vertices.
C'H.4 PTER .3. DIRECTED G R4PI-I P-4CKINGS
Figure 3.5: Gadget for t h e COL-{el-factor
Construction: Consider the vertex set V ( W )u X
problem
u V(Y)and
the arc set con-
taining t h e arcs from sets IY and Y. For each triple (.!, t',. t2, Isti) of T , add
a copy of t h e gadget t o the graph by identifying connector arc alas with tit{,
connector vertex b with t2, and connector arc clc2 with t&.
The interior arc of
these copies of t h e gadget make u p the arc set of the graph. All interior vertices
of t h e gadgets a r e added to the vertex set. Call the resulting graph G'.
Claim 3.2.2. The set T has a 3-dimensional matching if and only i j G has a
factor.
If 7' has a 3-dimensional matching, then for triples in the matching,
we have t h e COL-{el-factor of the corresponding copy of t h e gadget
shown in Figure 3.6.
Figure 3.6: A ~ o L - { f i ) - f a c t o r of the gadget including the connectors
For triples not in the matching, t h e corresponding copy of t h e gadget
has a
COL-(6
}-factor
as shown in Figure 3.7.
C'H.4 PTER .?. DIRECTED C R-4 PH P.4Cl\'INCS
Figure 3.7: -4 ~ ~ ~ { f i ) - f a c of
t o trh e gadget excluding the connectors
Thus, for any 3-dimensional matching of 7, we have a
COL-{el-
factor of G.
Suppose t h a t we have a COL-{fi j-factor of G. Consider the vertex
I
of a given copy of the gadget. If I is covered by a fi which contains the
vertes t b u t not t h e vertes w , then
must be covered either by the
fi
LL:
cannot be covered. Therefore, x
between w and
2.
o r by t h e
b t o z and then t o y. Suppose z is covered by the
Then y can only be covered if t h e
p4from z
fi
fi
from
from w t o z.
t o y is chosen. In this
case, none of the connectors a r e chosen and so the triple (a,6 , c) is not
in t h e 3-dimensional matching of 7.
If
I
is covered by the
by choosing the
fi
fi; from b t o z t o y, then w can only be covered
from w t o a for t h e packing and then r can only
be covered by choosing the
fi
from r t o c. In this case, all connectors
will b e covered and so we choose t h e triple (a,b, c) to be in t h e 3dimensional matching of T .
This is a 3-dimensional matching of T since it is generated from a
{el-factor.
T h e COL-{el-factor
problem is NP-complete.
Now t h a t we have shown t h e ~ o L - { f i ] - f a c t o r problem t o be NP-complete, we can prove
the NP-completeness of the
16,a } - f a c t o r
problem by reduction of t h e COL-{GI-factor
problem.
Theorem 3.2.4. The Restricted
{pl,F..)-/act~r problem
is NP-complete.
C1H.4PTER3. DIRECTED GRAPH PAC'IilNGS
Pmof: We will reduce
COL-{fi}
to t h e
S u p p o s e G is a n instance of
restricted
{PI,l?,}-factor
restricted
{fi,PI}-factor
problem.
COL-{el.
W e will c r e a t e a n instance G' o f t h e
problem in t h e following way: Firstly, remove edges
h a v i n g b o t h e n d vertices coloured 1 since these edges a r e not used in t h e
{ f i .G } -
f a c t o r problem. All o t h e r edges b e c o m e t h e edge s e t of G'. All vertices with
colour 1 make u p t h e independent set D and t h o s e with colour 0 m a k e u p t h e s e t
COL-{fi}.
t h e s e t D m u s t have size
A of VfG'). If we have a Y E S i n s t a n c e o f
3P a n d t h e s e t A must have size 2k, a n d all fi's will have vertices from D a n d
A a s shown in Figure 3.8. Since G a d m i t s a
{ e l - f a c t o r . T h i s implies t h a t G' a d m i t s a
COL-{fi j-factor, G'
must admit a
{PI,l?4)-factor.
Figure 3.8: P l a c e m e n t of
fi
Now, s u p p o s e G' is a Y E S i n s t a n c e o f t h e restricted
{fi,6 ) - f a c t o r
problem.
T h e n by Proposition 3.2.1, t h e r e is a {&}-factor F of G'. If we colour vertices
t.1,
u~ a m d us of each
0 , t h e n we have a
Corollary 3.2.1. The
P-f:
of t h e
fi
of F with c o l o u r 1 a n d vertices
and
"4
with colour
factor of G'. 0
{Fl,l?4)-factor problem
is A'P-complele.
A n y instance of t h e restricted {fi,fi}-factor
{PI,fi}-factor
problem is also a n instance
problem since b o t h probiems ask t h e s a m e question.
Following a similar approach to t h a t used for t h e
shown t h a t all
02
(6,
e}-factor
{fi,fi}-factor
0
problem, i t c a n b e
problems w h e r e j is even a n d greater o r equal to 4, a r e
N P - c o m p l e t e . T h i s again follows from ideas used in [41].
Infinite Families of Paths
S t e m m i n g from t h e g a d g e t for t h e {fi}-packing
for t h e
problem, a s well as from gadgets c o n s t r u c t e d
I&}-, {fi,A}-,{fi,a}-,
and ( f i ,fi}-pa~king problems individually,
we have been
C H A P T E R 3. DIRECTED GR4PH P4CKINGS
able t o construct a gadget which can be used t o prove t h e JVP-completeness of families of
directed p a t h s
family {
f i with k 2 2.
fi,&}.
Interestingly, o u r gadget works for all such families except the
However, t h e different approach t a b n in [12] allows this case to be treated
a n d i t is found t o be N P - c o m p l e t e as well. O u r gadget is constructecl in the following way:
Construction: Consider a family G of directed paths. Let pk be t h e shortest p a t h in
$! G, where 2k - 1 2 I 1 k + 2. T h e n we construct t h e
t h a t family. Let 8 E G a n d
gadget shown in Figure 3.9.
Figure 3.9: Gadget for p a t h factor problems
T h e connector b is a directed p a t h whose length is 2k
- 1 - 1.
T h e inequality 21;- 1 2 1
ensures t h a t t h e connector b h a s at least one vertex. T h e inequality 1
4
there is at least one vertex between t h e Pks x1
not work for t h e
{fi,fi}-pcking
. ..xk+l
> k + 2 ensures t h a t
a n d yl ...yk+l. This gadget does
problem since t h e inequality for this case gives 3 2 3
> 4.
We have t h e following theorem:
Theorem 3.2.5. The G-factor problem is .UP-complete for
G
*
{ f i ,. . ., 4, . ..}, where
2/;-111>k+2,!:22,8~~ondP;+~$~.
Proof: We will reduce 3DM to this G-factor problem. Consider t h e gadget of
Figure 3.9 with connector vertices a a n d c a n d connector b a p a t h of length
2k - I - 1. Consider t h e 3-dimensional matching problem with W and Y, two
s e t s having q points, a n d X , a set having q elements, each isomorphic t o a path
C H A P T E R .3. DIRECTED GRAPH PACKINGS
of length 2k
- I - 1.
consist of t from
41
For t h i s proof, we consider triples o f instances of 3DM to
W ,t g from 1' a n d
a p a t h t 2 , . . . . t 22k-l from X.
W e construct t h e g r a p h G as before on t h e vertex set WU-Y UY. For each triple
we a d d t h e arcs a n d interior vertices of a copy of t h e gadget t o t h e g r a p h
of 7,
by associating a with t l , b with t h e
-
PZk-I-tt
t Z , , . . .,t22k-r, a n d c with t s .
Claim 3.2.3. 7 has a 3-dimensional matching if and only if G has a G-factor.
P m f : Suppose t h a t 'T has a 3-dimensional matching.
Then for
triples in t h e matching, we have a G-factor of t h e corresponding copies
of t h e g a d g e t s as s h o w n in Figure 3.10.
Figure 3.10: A G-Factor including t h e connectors
For those triples n o t in t h e matching, we have a G-factor of t h e corresponding copies of t h e g a d g e t s as shown in Figure 3.11.
Figure 3.11: A G-Factor excluding t h e connectors
C H A P T E R .3. DIRECTED GRAPH P,.\CfiliVCS
Thus, if we have a 3-dimensional m a t c h i r ~ gof T , t h e n we have a
G-
factor of t h e g r a p h G.
Suppose t h a t we have a 6-factor of C'. Consider o n e of t h e copies of the
gadget in G. Vertex
by t h e
c a n either be covered by t h e
o r by t h e Pk
zl
4 X I . . rl+l,
XI
... t k a .
f i 1 1 .. . I ~ I ~ + I ,
Suppose t h a t it is covered
by t h e first. T h e n we c a n n o t use t h e connector 6. Therefore, t h e
vertices
path
tk+2,
...,21+1
-
c a n n o t be covered by a. f i o r by a fl since t h e
h a s length less t h a n k. T h u s s u p p o s e t h a t
Xk+2. -.xl+,
6
choose the Pk . ..
covered by t h e
X I . ..21+1.
yl
yk+l.
T h e n in order t o cover
yk+l,
xl
is
we must
In this case, none of t h e connectors a r e in
t h e G-factor a n d so t h e triple ( a , b, c ) is not p a r t of t h e 3-dimensional
matching of T.
4
Suppose t h a t xl is covered b y t h e Pkx l . . .xka. T h e n since xk+l
and
xk+l
.. .xl+l yk+l
. ..zr+l
a r e b o t h p a t h s of length less t h a n k, t h e oniy
-
way t o cover z k + l is t o choose t h e Pk bl ...b 2 k - 1 x k + l . . .x l + l y k + l - T h i s
then forces us t o choose t h e
yl
. ..yk.
6yl . ..ykc in order t o cover t h e vertices
In this case, all connectors a r e covered a n d so ( a , b, c) is p a r t
of t h e 3-dimensional matching of T.
Thus, if we have a G-factor of G, we have a 3-dimensional matching
of T . 0
T h e F-factor problem for
GC
G I4 4 , is N P - c o m p l e t e .
3.3
{a..- 8- ..I, 2k -
12 1
2 k + 2 , k 2 2,
fi E G,
0
Main Theorem
T h i s theorem makes use of augmenting configurations. A n augmenting conjigurntion of a
g r a p h H with respect t o a packing P of H is a special subgraph of H such t h a t t h e number of
vertices covered by
edges from
P in t h e s u b g r a p h c a n
be increased, o r augmented, by removing certain
P a n d adding o t h e r edges to P , o r simply by adding edges t o P. RecaIl t h a t an
a r c ub is a directed edge beginning at v e r t e s a a n d ending in vertex 6. If we d o n o t know
t h e direction of a n a r c o r if t h e direction is n o t i m p o r t a n t we will simply refer t o it as t h e
edge a b . A walk is a sequence
voelvle2
.. .e k v k of vertices a n d edges such t h a t e;
= v;- 1 v;
CH.4PTER 3. DIRECTED GRAPH PACI\'ING'S
for all i. If no edges of t h e sequence a r e repeated. then we have a tmil. We will simply give
t h e vertices of t h e trail in this proof. T h e r e a r e directed edges between consecutive vertices
of t h e trail. A11 of o u r augmenting configurations a r e trails. N o t e t h a t a vertes may b e
repeated in a trail. In this case, t h e augmenting configuration is set'j-intersecting,
Theorem 3.3.1 s t a t e s o u r main result: a
{fi,fiJpacking
is maximum if and only if there
a r e n o augmenting configurations. A polynomial time algorithm for t h e { P I ,&)-packing
problem c a n b e found in [12]. T h i s algorithm is inspired by Hall's theorem and the resulting
algorithm for matchings in bipartite graphs. In the remainder o f this section, t h e t e r m
packing will be used t o refer to a
16,fi)-packing.
In this chapter, a n alternating tmil is a trail
UO, VO, U,,v1,
- . . vk-1,
uk with a n even
number of edges, where uo is t h e s t a r t vertex, a n exposed v e r t e s of P . For each i f
{O? 1,. . ., k
- 1) we have t h e following:
a u;v; is a n edge of
H which is n o t in t h e packing P .
a q u , + ~is a n edge of t h e packing
a T h e edges u i v i and v;u;+l
P.
a r e either both oriented t o w a r d s t h e vertex
tl;
o r both
oriented away from t h e vertes u;.
a u;+l is incident with exactly o n e a r c of
P a n d v, is incident with one o r two a r c s of P.
Note t h a t if u; is incident with two arcs of P , then o n e of t h o s e arcs is directed towards
v; a n d o n e is directed away from v;. Also, n o vertex u; can a p p e a r twice in a n alternating
trail.
An augmenting configuration is a n alternating trail of one of t h e following types:
1 . T y p e 1: A trail U O , VO,U I ,v1 . . ., Z L 2,
~ , where uo, vo, u1,
a n d z is a n exposed vertex of
2. T y p e 2: A trail uo, v0, ul,~1
. . ., uk is a n alternating trail T
P.
...,uk,Z, where uoTt~o,ul?..-,uk is a n alternating trail
T
and
(a) r = o;, 0
p a r t of a
(b) z = u;, 0
p a r t of a
5 i 5 k - 1. T h e n
ukviu; is a
p2 but is n o t
in P a n d v;u;+l in
P is n o t
p2 of P, o r
5 i 5 k - 1. T h e n
f32 of P.
UkUiVi
is a
p2 b u t is n o t in P
a n d v i u i + ~in
P is n o t
C'M.4 P T E R -3. DIRECTED GR4 Pff P.-\C'I\'fNC;S
- .., u k , y. z . where rro. ~'0.u l . . ..: u k is a n alternating trail.
is not in P and yz is in P.
3. T y p e 3: A trail uo,
uky
UO, U I ,vl
-
(a) u k y z is a P2. o r
(b) z = v i 7 0 <_ i
5 k - I.
Then
yviui+l
is a
f3 in P.
4 . T y p e 4: A trail uo, vo, u l , v l . .., uk,x, y, Z, where uo. uo, u l , . . . ?
trail.
u p
is not in P and zyr is a
uk
is a n alternating
fi of P.
T h e augmenting configurations of type 1: 3a,and 3 are not self-intersecting. Recall t h a t
this means no vertex is encountered more t h a n once in the trail. Augmenting configurations
of type 2a, 2b and 3b are self-intersecting. Esamples of each type of augmenting configuration are given in Figure 3.12. In this figure, dotted lines are edges which are in t h e packing
P and solid lines are edges which are not in P . Circled vertices are exposed in P .
In all of these types of augmenting configurations, .uo is a n exposed vertex which we call
the s t a r t component, any sequence of vertices
y and
2
u;viui+l
is called a middle component and x ,
a r e end components. These components are represented in Figure 3.13.
C'H.4 PTER 3. DIRECTED GRAPH P.4 C,'i\'liVGS
Configuration
Example
Type 1
"1
Type 4
"I1
U
2
X
Y
----------).
Figure 3.12: Examples of augmenting configurations for the
z
{ f i , p2]-packing problem
C'H-4PTER -3. DIRECTED GRAPH PACKINCS
Start components
Middle components
End components
* denotes that this vertex may have been previously encountered
Figure 3.13: Augmenting configuration components for the
{fi. f i ~ - ~ a c k iproblem
n~
C'M.4 PTER .3. DIRECTED GRAPH PACKIXGS
Now we will look at how augmenting configurations a r e augmented. Augmentecl config-
urations for each e s a m p l e of Figure 3-12 a r e given in Figure 3.14.
Configuration
Augmented example
Figure 3.14: Examples of augmented configurations for t h e
{PI,&}-packing
problem
Consider t h e type 1 configuration. T o augment, remove edges v;u;+l,i = 0, . ..k - 1 from
P and add edges u p ; , i = 0,.. .,k - 1 a n d ukz to P . For t h e type 2a a n d 2b configurations.
remove edges v;u;+l,i = 0, ...k - 1 from P and a d d edges u;v;, i = 0 , . . .,k - 1 a n d ukr t o P .
For a type 3a configuration, remove edges viui+l, i = 0, .. .k - 1 from P and add edges uivi,
i = 0 , . ..,k - 1 , uky to P a n d keep edge yz in P. To a u g m e n t t h e type 3b configuration,
remove edges viui+l, i = 0 , . . .k - 1 and yz from P and add edges u i u i , i = 0 , . . ., k - 1 a n d
C H A P T E R 3. DIRECTED GRAPH P.4C'KfNG'S
trk. t o P. FinaIly. t o augment type 4 configurations. remove edges t 7 , u ; + l . i = 0 , . . .A- - 1
- 1 and ukx t o P and keep edge g z in P.
tn Theorem 3.3.1, we will use t h e notation G \ ab, where ab E E ( G ) t o represent t h e
subgraph of G having the same vertex set as G and with edge set E ( G \ ab) = E ( G ) \ a l .
and xy from P. add edges
Theorem 3.3.1. A
uiv;,
i = 0 , . ,..k
{PI.
6)-pucking P
o/ o directed gmph G is of maximum size if and
only if there exists no augmenting configurntion.
Proof: Suppose G contains an augmenting configuration with respect to P.
Then clearly by augmenting we will increase the size of P and s o i t could not
have been of maximum sizeSuppose that P is not of maximum size. Let P' be a larger
{fi, fi)-packing
of
G. We will prove t h a t P must a d m i t a n augmenting configuration.
We prove the following statement by induction on IE(G)I! the number of edges
in the graph G:
For all k , if G has k more vertices covered by P' than by P , then there are k
vertices which start an augmenting conjgunrtion in G with respect to P .
Consider the case I E ( G )I = 0. In this case, k = 0 and s o we have 0 augmenting
configurations and the statement holds.
Suppose t h a t t h e statement holds for all k for any graph with fewer than J E(G')l
edges.
Suppose t h a t G has
{PI,& ~ - ~ a c k i nP~ sa n d P'
such t h a t there are
k 2 1 more vertices covered by P' than by P. We will use the terminology
P-edge and P'-edge t o denote t h a t a n edge is a 6 of P , o r of P', respectively.
Similarly P-path and PI-path. will denote a p2 which is in the packing P , o r
of P' respectively. Note we can assume E ( G ) = P u PI. If not, consider the
spanning subgraph of G with edge set P U P ' and apply our statement. T h e k
vertices which begin augmenting configurations in this spanning subgraph also
begin augmenting configurations in G. Let v be a vertex covered only by P'.
Then we have the following three cases t o consider:
1. T h e vertex v is covered by a Pt-edge v w .
2. T h e vertex v is covered by a P'-path vwu.
3. T h e vertex v is covered by a P'-path wlvwz.
CHAPTER 3. Dl RECTED GRAPH PAC,'IiI!VGS
We will now t r e a t each case with i t s accompanying subcases. All these proofs
follow a similar format. We will look a t a s u b g r a p h G\ab of G ancl we will prove
t h a t k vertices s t a r t augmenting configurations of P restricted t o G'\ab a n d these
a r e either preserved o r c a n be modified t o produce augmenting configurations
in G. Note t h a t in t h e following proofs. t h e term augmenting configuration will
always refer t o a n augmenting configuration with respect t o t h e packing P .
1. Suppose vertex u is covered by a Pr-edge cw.
( a ) S u p p o s e u: is also esposed in
packings P a n d
P'
restricted
P. Consider
t o G \ uw. In
the graph G \ vw and the
this g r a p h , there a r e k - 2
vertices covered by P' a n d n o t b y P a n d so by o u r s t a t e m e n t , there a r e
k
- 2 vertices which
begin a u g m e n t i n g configurations in G \ v w . Since
v a n d w a r e exposed with respect t o b o t h
P
a n d P' in G \ urn, they
c a n n o t a p p e a r in any augmenting configurations. Consequently, none
of t h e k - 2 vertices beginning augmenting configurations a r e affected
by t h e edge uw a n d so all augmenting configurations a r e preserved
in G . Clearly, b o t h v a n d
u;
begin augmenting configurations in G.
namely vw a n d w v , a n d so we have k vertices which begin augmenting
configurations in G.
(b) S u p p o s e w is covered by a P - e d g e wu a n d w u @ P'. Consider t h e g r a p h
G \ wu. In this graph, P' covers k + 2 more vertices t h a n P. Vertices v
a n d w begin augmenting configurations, v w a n d wv, in G \ w u . Since
neither u nor w a r e incident with a n y P-edges o r P - p a t h s in G \ wu,
these a r e t h e only two a u g m e n t i n g configurations in which they a p
pear. Therefore, there a r e k o t h e r vertices which begin augmenting
configurations a n d we need only consider those augmenting configurations which contain u since all augmenting configurations which d o n o t
involve u a r e augmenting configurations of P in G as well. T h e exposed vertex u must either s t a r t o r e n d a n augmenting configuration.
S u p p o s e there is a n augmenting configuration of t y p e 1 U
where z = u. If ukuw is a
4, t h e n
.. .ukz,
~ V O U ~
u o v o . . . U ~ U Wis a n augmenting
configuration of t y p e 3a in G . Otherwise, ukuw is a middle component
a n d t h e n uovo.. .u ~ u w vis a n augmenting configuration of type 1 in
C'MAPTER 3. DIRECTED GRAPH P-4CKINGS
G. At this point, w e know t h a t any v e r t e s uo # u which s t a r t s a n
augmenting configuration of P in C \ tuu which does n o t contain u o r
which ends a t u also s t a r t s a n augmenting configuration in G. We also
know t h a t there a r e at least k - 1 such vertices. If t h e r e a r e El., then
t h e proof is complete. If not, then t h e k t h s t a r t vertes m u s t be u .
Suppose u begins a n augmenting configuration
y r , o r z y z , in G
\
mu.
If vwu is a
6, then
..
U V ~ U ~ u. k E t
E = z,
u w u is a n a u g m e n t i n g
configuration of t y p e 3a s t a r t i n g a t v in C. Otherwise, u w u v o . . .ukE
is a n augmenting configuration in G. T h i s gives the k t h vertex which
begins a n a u g m e n t i n g configuration in G .
(c) Suppose t h a t w is o n a P - p a t h wuu'. Consider t h e g r a p h G \ wu. In
this graph, P' covers k
+ 1 more vertices t h a n P. As before,
vw a n d
wv a r e augmenting configurations so we will t u r n our a t t e n t i o n t o t h e
remaining k - 1 vertices? o t h e r t h a n u a n d w t which begin a u g m e n t i n g
configurations in G
\
wu.
Since u a n d u' a r e covered by a P - e d g e
in G \ mu, neither c a n s t a r t a configuration. Consider a n a u g m e n t i n g
configuration which begins a t a v e r t e s uo. Suppose t h a t uu' is a P-edge
in this configuration. Note t h a t u a n d u' cannot appear on a P-edge
o t h e r than uu' in a n a u g m e n t i n g configuration since uu' belongs t o t h e
P - p a t h wuu' in G a n d this would imply t h a t one of them was incident
with three P-edges in G o r was on a p a t h of
P of length three. Since
t h e argument for s u c h a configuration, with uu' a P-edge in C\ wu a n d
p a r t of a P - p a t h i n G, will b e used frequently throughout t h e proof
of Theorem 3.3.1, w e wilI prove t h e following lemma t o which we will
refer throughout t h e remainder of t h e proof.
Lemma 3.3.1. Let G be o graph, P a
{PI,6 ) - p a c k i n g of G and let a6
be a P-edge of some P-path. If G \ a6 has an augmenling configumtion
of P beginning at uo, uo # a,6, then either there is an augmenting configuration beginning at uo in G o r there is an augmenting configumtion
of type 2a wilh a = z and b not in the configumtion.
Pfoof: S u p p o s e a is t h e middle vertex of t h e P - p a t h a'ab,
otherwise reverse t h e roles of a a n d 6 . In t h e graph G \ ab, if b
is a n exposed vertex in an augmenting configuration beginning
CN-4PTER 3. DIRECTED GR.4PH P'4C.'IiINGS
at a vertex uo, which does not contain a' o r a. then b = 2. a n d
so in G we woiild have a n augmenting configuration uo .. .baa'
of t y p e 4.
If v e r t e s a is covered by some P - e d g e a'a in some augmenting
configuration uouortl . . .vk-1 ukE in G \ ab, where E = z, y x o r
xyz, then either b is in t h e configuration o r i t is not. If 6 is
in t h e configuration, then b = z a n d we have an augmenting
configl~rationof P in G of t y p e 3b with a = v,. and a' = uj+r.
ff b is not in the configuration, then a is a u ; , 0 < i 5 I; o r a v ; ,
0 5 i < k. If a = u; in a n y t y p e of configuration, then a' = u;-1
a n d in G we have a type 4 ending a'ab t o t h e configuration.
I t remains t o consider t h e case where a = vi- If a = u; a n d in
t h e trail from uo t o a no vertex is revisited in t h e augmenting
configuration, t h a t is, we have configurations of type 1, 3a o r 4
o r we a r e in t h e p a r t of configurations 2a, 2b a n d 3b between uo
a n d t h e revisited vertex, then a' = u;+l. a n d in G t h e P-edge
a6 is p a r t of a middle of this a u g m e n t i n g configuration a n d so
t h e augmenting configuration is preserved in G'.
Now suppose a = v; a n d t h e vertex u, o r uj, j
5 i is revisited
in
t h e augmenting configuration. T h e possible configurations a r e
those of types 2a, 2b a n d 3b, First, consider a n augmenting
configuration of t y p e 2a. Either i = j o r i
>
j. If a = v ; ,
a n d i = j, then we have the second result of the lemma. If
a = v;, a n d i
> j, then
in
G this a u g m e n t i n g configuration will
be preserved, t h e P-edge a b will j u s t b e p a r t of a middle.
Suppose t h a t t h e atlgmenting configuration uouo ... ukz is of
t y p e 2b. T h e n vertex uj is revisited a n d for a = v;, we have
2'
>j
a n d a' = ui+l in t h e alternating trail uovo.. .uk. In G
w e c a n find a new augmenting configuration beginning at uo
in t h e following way. Travel along t h e alternating trail from
uo to uj.
Instead of continuing to u j , follow t h e edge ujuk.
If ujurvk-1 is a
4, then
we have encountered a n augmenting
configuration uovo ...U j U k V k -
1
of t y p e 3a. Otherwise, we have
C'HA PTER 3. DIRECTED GRAPH PACYiINGS
a middle u, ukuk-1- In this case, if
vk-1
is t h e middle vertex of
a P - p a t h . then we have a n ending of type 4. If not, we have
a middle -ujukvx.,l a n d we look a t the nest edge covered in
P , u ~ , ~ u c ; - Lo n the aIternating trail. Again, we will have a n
ending of t y p e 3a, o r of type 4 o r we will have a middle. If
we have a middle, continue t o t h e next edge. We u 4 l find a n
ending since t h e vertex a = v; of t h e edge aa' is t h e middle
vertex of t h e P - p a t h a'ab.
Finally, suppose t h a t a n augmenting configuration of type 3b
is found in G \ ab. Vertex a could never equal
uj,
where
uj
is
t h e revisited vertex, since this would mean t h a t a was incident
i > j,
a n d in G this augmenting configuration is preserved since a6
with three P-edges in G. Thus, a = v; and a' =
u;+l,
will just be p a r t of a middle of t h e augmenting configuration.
0
We now return t o case l c where w is o n a P - p a t h wuu'. Note t h a t
for a configuration of type 2a in G \ w u , if u = v; = z , then in G
we have the augmenting configuration u o ~.-.u , u t . ..U W U . Augment
this configuration a s follows: -411 arcs of t h e augmenting configuration
P a n d uncovered in P ' ,
a n d all arcs of t h e augmenting configuration which were covered in P
which were covered in P' become covered in
become covered in
P' a n d
uncovered in P.
.411 augmenting configurations of P found in G \ wu a r e preserved o r
c a n b e modified t o produce new augmenting configurations, beginning
at t h e s a m e vertex uo, in G, as shown in Lemma 3.3.1. Therefore,
there a r e k
-I
vertices which begin augmenting configurations in G
a n d which d o not begin at
t. o r
w. T h e augmenting configuration vwuu'
gives us k vertices which begin augmenting configurations in G.
(d) Suppose w is t h e middle vertex of a P - p a t h
U'WU,
oriented from u' to
u. Either vw a n d u'w form a middle component o r vw and uw form a
middle component. Say i t is vw a n d u'w, otherwise reverse t h e name
of u a n d u'. Consider t h e graph G \ U'W. Since t h e vertex u' is exposed
in G \ u'w,
P' covers k + 1 more vertices than P . Therefore there a r e
C'HAPTER 3. DIRECTED G'RA PH PACl\'lhrGS
a t least k vertices other than u' which s t a r t augmenting configurations
in
C \ ulw. By Lemma 3.3.1 ail a r e preserved in G except possibly a
type 2a configuration with w =
vjjz
and u =
uj+l.
However. such a
configuration cannot exist since this would mean t h a t w was incident
with three PI-edges in G.
2. Suppose vertex v is covered by a P'-path vwu.
( a ) Suppose w is exposed in P and consider t h e graph G' \ v u;. The packing
P' covers k - I more vertices t h a n P in G \ v,w and so there are k - 1
vertices which begin augmenting configurations in G \ cw and all of
these configurations are also configurations in G. T h e reason for this
is t h a t adding t h e PI-edge vw to any augmenting configuration cannot
destroy the configuration since w is exposed in
C \ vw. Since vw is
a n augmenting configuration of G beginning a t v, we have k vertices
which begin augmenting configurations in G.
(b) Suppose t h a t w and u a r e covered in P by (i) the P-edge w u , o r (ii)
a P-path wuu', o r (iii) a P - p a t h u'wu.
Examples of these cases a r e
shown in Figure 3.15. Note t h a t in these diagrams, t h e edge w u is in
both P and P'.
(0
(ii)
(iii)
Figure 3.1.5: Example diagrams for case '2b
For cases (i) and (ii), consider t h e graph
G \ vw. In this graph, P'
covers k - 1 more vertices than P. Since no vertices covered in P have
become exposed in P through t h e restriction t o G \ v w , any augmenting
configuration beginning a t uo involving zu, u o r u' in G \ vw is still a n
augmenting configuration beginning at uo in G since adding P'-edges
cannot destroy augmenting configurations.
In case (i), vwu is a n augmenting configuration of type 3a in G, and in
CHA PTER :3. DIRECTED GrW PH PACKINGS
case (ii), uwurr' is an augmenting configuration of type 4 in G and so in
both cases, there are k vertices which begin augmenting configurations
in G.
For case (iii), consider t h e graph G
k
+ 1 more vertices than
P in G' \
wu'.
\
wu'.
P. T h e vertex
u'
In this graph. 13' covers
is esposed with respect t o
If ut begins a n augmenting configuration, then in G,
this augmenting configuration will b e destroyed. Therefore, there a r e
k other vertices which begin augmenting configurations in G \ u:uf. T h e
rest of this proof follows from Lemma 3.3.1. Note t h a t we cannot have
a n augmenting configuration of type 2a with w = r and u = u j + l since
this would imply t h a t w had degree three with respect t o P' in G. T h u s
there a r e 12 vertices which begin augmenting configurations in G.
(c) Suppose t h a t wu is not in P . Then w can be covered by (i) a P-edge
aw, or (ii) a P-path a'aw, o r (iii) a P - p a t h awu'. Examples of these
cases a r e shown in Figure 3.16.
(ii)
(i)
(iii)
Figure 3.16: Example diagrams for case 2c
In cases (i) and (ii), consider the graph G \ aw. In case (iii), if vwa' is a
6,consider G \ aw. Otherwise, v w a is a fi and then consider G \ a t u .
In case (i), P' covers k + 2 more vertices t h a n P. In cases (ii) and (iii),
Pr covers k
+ 1 more vertices than P. In cases (i)
and (ii), vw and
wv a r e augmenting configurations and s o there a r e k and k
-I
vertices respectively which begin augmenting configurations in
other
G \ aw.
Consider case (i). If an augmenting configuration contains a, then i t
either begins o r ends in a. If it begins at a, then we can construct a n
augmenting configuration beginning at v as we did in case l b . Suppose
a n augmenting configuration begins a t some uo
#
a and ends a t a
CHAPTER .3. DIRECTED GRAPH P-4CIilhrGS
-
with t h e edge u k n . If uka a n d aw form a P2. then uo . . .u k a w is a n
augmenting configuration of t y p e 3a in G'. Otherwise. a w u provides a n
ending of type 1 for t h e augmenting configuration in G.
Suppose a n augmenting configuration e n d s at v e r t e s w. T h e n it must
be a n augmenting configuration uovo .. . urn o f t y p e 1. If uwa is a I%'2.
then uua is a t y p e 3a configuration in G. Otherwise, uwa must form a
middle. In t h i s case, we consider vertex a which either begins a n augmenting configuration o r does not. If a begins a n augmenting configuration a
...u kE, then consider uot.0 . . . uw a n d a . . .ukE . where a # uo.
If
no vertex of a ...ukE a p p e a r s in uovo.. .uw. t h e n uovo.. .u w a . . .ukE
is a n augmenting configuration of G. If a v e r t e s is repeated, then it is
a vi, 0
5 i < k, o r
< i 5 k. If vertex u; is t h e first vertex of
uovo . . .uw, then uo .. .wwa . ..v; is an aug-
a u;, 0
a .. .u k E which is also in
menting configuration of t y p e 2 a in G provided t h a t v is n o t t h e middle
vertex of a P - p a t h. If vi is t h e middle v e r t e s of a P - p a t h, then consider
uo .- .t'; - ..w a .. .v; . . .u k E in G . First a u g m e n t between v; and E by
switching along t h e p a t h v; . .- uk and augmenting t h e ending E. Then
uo
.. .u; . . - wa.. .u;
is a n augmenting configuration of t y p e 2a in C. If
a vertex u; is t h e first vertex of a .. . ukE which is also in uovo . . .uw,
then uo . . .uwa ...u ; is a n augmenting configuration of t y p e 2b in G.
Now consider uovo. ..u w a n d a . ..ukE . where a = uo. T h e n there is
one augmenting configuration which begins at n a n d o n e which ends at
a and
ends at w. T h e n we c a n construct a n augmenting configuration as in l b
w . This is simply a n augmenting configuration which begins a t
and we preserve o u r k vertices which s t a r t augmenting configurations.
Now suppose we have a n augmenting configuration uovo .. .uw of type
I, uwa is a middle, and a does n o t begin a n augmenting configuration
in G
\ aw. We
claim t h a t i t is possible t o modify so t h a t only one
augmenting configuration e n d s a t w. T o see this, suppose t h a t two
augmenting configurations e n d at w, o n e beginning at a vertex uo and
t h e other at a vertex c. Both must e n d with t h e Pf-edge
UW.
NOW
either u is t h e first vertex common t o both augmenting configurations
o r it is not. If i t is, t h e n t h e augmenting configuration beginning at uo
C'H.4 PTER 3. DIRECTED GRAPH P=IC'f\'IXC;S
has a P-edge 11 u and the augnienting configuration beginning at c has
a P-edge l z u . T h e edges 1 , u and 12u must form a
p2 and so one of the
augmenting configurations, say the one beginning at c can be modified
t o an augmenting configuration c .. .lzul 1 of t y p e 4. Then there is just
one augmenting configuration ending a t w.
If u is not t h e first intersection of the two augmenting configurations,
then the first intersection is at some vertex vi, 0
5 i < k, o r
ui?
0 < i 5 k- Note that if vo is t h e first intersection, the uouoc is a P'4
path and so either uovoul o r maul is a
P2 and hence a n augmenting
configuration of type 3a which does not end a t w. If the augmenting
configurations first intersect at a u;, then either l l u i is a P-edge of
u o . . . l l u ; . . . w and 12ui is a P-edge of c . . . l z u i . . .w, o r ll.ui is a
edge and f2u; is a P'-edge.
P-
In the first case, l l u i f z is a P - p a t h a n d
so one augmenting configuration, say the one beginning a t c can be
modified t o become the augmenting configuration c.. .12.ull of type 4 ,
and the augmenting configuration beginning at uo and ending at .w is
preserved. In t h e second case, we have lzuit.; which is a Pf-path, not
a middle and so the augmenting configuration beginning a t c is not
legitimate.
If the augmenting configurations first intersect at some vertex .v;,then
either l l v ; and 12vi are PI-edges o r else llui is a P'-edge a n d 12vi is
a P-edge. In t h e first case, either l l ~ i ~ ior
+ l1 2 ~ i ~ i + is
l a F$ and
s o the corresponding augmenting configuration is actuaiiy of type 3a
with ending f l ~ i ~ i +o rl 1 2 ~ i u ; +and
l
s o there is only one augmenting
d
configuration ending at w. In t h e second case, f 2 v ; ~ i +must
1
be a P2
and s o c . . . 1 2 u ; ~ i +isl a n augmenting configuration of type 4 a n d only
one augmenting configuration ends at w.
Therefore, only one augmenting configuration ends at w. !f uwa is
a
fi, then
this augmenting configuration acquires the ending uwa of
type 3a in G. Otherwise, this augmenting configuration is destroyed
by the addition of aw. However, in this case, vwa is a n augmenting
configuration beginning a t vertex u and so we still have k vertices which
begin augmenting configurations in G and we are done.
C'H.4 PTER 3. DIRECTED GR4 PN P.4CIiIArGS
Case (ii) follows from Lemma 3.3.1. In t h e case of a n augmenting
configuration of type 'La with a = u, = z , a n d a' = rr,+,.
menting configuration is destroyed in
C' but vtcaa'
t h e aug-
is a n augmenting
configuration of G' a n d so we still have k vertices beginning augmenting
configurations.
In case (iii)? if we a r e considering C \ a w , t h e n vwa' is a n augmenting
configuration of C . Otherwise, vwa is a n augmenting configuration of
G\af w . In either case, there a r e k o t h e r vertices which s t a r t augmenting
configurations in G
\ aw,
or G
\
a'w.
Case (iii) then follows from
L e m m a 3.3.1. An augmenting configuration of t y p e 'La not involving
v with w = uj = r ? a n d a' = uj+l (or a
= u,+l if we have G \ d w ) ,
If t h e
augmenting configuration does s t a r t a t v , t h e n i t is destroyed in G a n d
we use t h e augmenting configuration vwa' ( o r vwa) instead.
c a n n o t exist since w c a n n o t b e incident with three PI-edges.
3. Suppose vertex o is covered by a P f - p a t h w,vwz, oriented from
If either
wl
wl
to
ZC~.
o r w2 is exposed in P , proceed as in case 2a.
( a ) Suppose t h a t w l a l is a P-edge a n d t h a t t h e a r c
w2a2
is covered by (i)
t h e P-edge wza2, o r (ii) by t h e P - p a t h w2a2a',. These cases a r e shown
in Figure 3.17.
Figure 3.17: Example diagrams for c a s e 3a
Consider t h e graph G \ wza2. In case (i), t h e r e a r e k
+ 2 more vertices
covered by P' t h a n by P. In case (ii), this n u m b e r is k+ 1. In b o t h cases,
vwz and w2u a r e augmenting configurations and so there a r e k other
vertices in (i) a n d k - 1 in (ii) which s t a r t augmenting configurations
C'NA PTER 3. DIRECTED G R.4 PH P-4CId!VCS
in G\
w2a2.
T h e remainder of case (i) is analogous t o case I b and case
(ii) is analogous t o case lc. since w l a n d a l a r e n o t affected by the
removal of t h e edge
~ 2 ~ 2 .
If t h e direction of wlvwz is reversed, then case (i) is t h e s a m e as case
3b (i) a n d case (ii) is unaffected.
(b) Finally, suppose vwza2 a n d uwlal a r e middles. T h e n (i)
62W2
is a P-
edge, or (ii) naw2ai is a P - p a t h . These cases a r e shown in Figure 3.18.
Figure 3.18: Example diagrams for case 3b
Consider t h e graph G \
a n d n o t by P is k
w2a2.
T h e number of vertices covered by
+ 2 in case (i) a n d I; + 1 in case (ii).
t h e augmenting configurations beginning at v a n d
w2
P'
If we disregard
in case (i) a n d
beginning a t v in case (ii), there remain A; o t h e r vertices which begin
augmenting configurations in both case (i) a n d case (ii).
Consider case (i). T h e vertex
a2
is exposed a n d so i t may s t a r t o r
end augmenting configurations. Suppose a n augmenting configuration
starts a t
a2.
T h e n vw2az s t a r t s this configuration in G. If we have a n
augmenting configuration uo .. .ukaz of t y p e 1 , then if
+
ur,a2w2
is a P2,
we have a n augmenting configuration of t y p e 3a in G. Otherwise, a2w2v
provides a new ending o f t y p e 1 in G. An augmenting configuration
cannot e n d a t
w2
since
.w2
is t h e end vertex of t h e P'-path
wlvw2
and
so n o o t h e r P'-edge c a n contain w.
For case (ii), see L e m m a 3.3.1. Again, we c a n n o t have a n augmenting
configuration of t y p e 2 a with a; = uj+l a n d
incident with t h e P'-path
wl
vwz in
G.
w2
= vj = 2, since
w2
is
CH.4PTER 3. DIRECTED GR-4PH PAC'IiIiVGS
If the direction of wl r w z is reversed. then case (i) is t h e s a m e as case 3 a
(i). For case (ii) , reverse t h e roles of a2 a n d a: and t h e case is identical.
In a11 cases, if there a r e k vertices of G' esposecl in
which begin augmenting configurations of
P in
P,then there a r e k vertices
G'. This completes t h e proof of
o u r theorem. D
3.4
Packing with Directed Cycles
In this section, we consider packing with one-element families of directed cycles. We will
n o t consider t h e { ~ i } - ~ a c k iproblem
n~
since it is polynomial time solvable analogously t o
the
{fi }-packing problem of Theorem 3.2.1.
The Family B = {G}
Theorem 3.4.1. T h e { c } - f a c t o r problem is JVP-complete.
Proof: This problem is in t h e class JVP since given a {c3}-factor. we can check
t h a t all vertices are covered by o n e a n d only one d j in polynomial time. In
order t o prove the {c3}-factor problem NP-complete, we will reduce 3DM t o
t h e {C3}-factor problem.
Consider t h e gadget of Figure 3.19 having connector vertices a',
a2,
a n d a3. We
construct a graph G in t h e following way. Let the vertex set of G be W U S U Y.
For each triple of 7- = { ( t l , t z , t 3 ) l t l E W.t2 E X , t S E Y ) , a d d a copy of t h e
gadget to t h e graph by identifying a1 with t l ,
a2
with t 2 a n d as with t 3 .
Claim 3.4.1. T h e set T has a .?-dimensional matching i j and only
il G
has a
{C3}
-factor.
P m f : Suppose T has a 3-dimensional matching. Then for each triple
in t h e matching we have t h e { G I - f a c t o r of Figure 3.20.
For each triple not in t h e matching, we have the {c3}-factor
of Fig-
u r e 3.21.
Clearly there is always a { G I - f a c t o r of G when there is s 3-dimensional
matching of T.
C'H.4PTER 3. DIRECTED GRAPH PA CKlNGS
Figure 3.19: Gadget for the {G}-factor problem
Figure 3.20: A {G)-factor containing the connector vertices
Suppose we have a {GI-factor of G. This part of t h e proof is similar
t o t h a t of Theorem 2.1.1, the undirected case of this problem, with
the exception t h a t here we have connectors a l , a;! and as in place of
a, b and c. The reader can see t h a t just as in Theorem 2.1.1, if a
connector, say
23
is to have
al,
is covered by a l x 2 x l , then t h e only way t o cover
233313
in the {c3)-factor and this in turn necessitates
having yla2y2 and rla3z2 in the {C3}-factor. In this case, we choose
the triple ( t l , t 2 ,t 3 ) of T t o be in t h e 3-dimensional matching. If a1
is covered by a
22,
63
other than zlal t z , then in order t o cover
tl
and
we must choose 2 1 2 2 Z 3 . Eventually, we find t h a t a2 and a3 must
C'I-I.4 PTER 3. DIRECTED GRAPH P.~CI<IIVGS
Figure 3.21: A {e3}-factor
excluding t h e connector vertices
also b e covered by c 3 ' s o t h e r t h a n yla2y2 a n d z ' a 3 z ~ in this case,
T t o be in t h e 3-dimensional
matching. T h i s produces a 3-dimensional matching of T . a
we d o n o t choose t h e triple (t
t z , t S ) of
problem is ,LrP-
T h u s , 3DM a {c3)-factor problem and so t h e {c;)-factor
complete. CI
The Family
E = {ck)
Theorem 3.4.2. Let k 2 3 be an integer. The {&}-factor problem is NP-complete.
Proof: A s
is in
NP.
was t h e case for t h e {c3}-factor problem. t h e (ck}-factor
problem
W e will give t h e construction of t h e g a d g e t using t h e case
c4as
o u r example. T h e proof of this theorem uses reduction of t h e k-dimensional
matching problem.
Construction: To construct a g a d g e t for this problem, we begin with a directed
having vertices yl,. ..,yk. To each of the vertices of this central cycle we
attach G ' s which a r e vertex-disjoint from each o t h e r . To each of these k
c7#'s
we a t t a c h a connector vertex a;,i = 1.. .k, using t w o directed edges. These
edges a r e incident with t h e two vertices on a n edge with a vertex y;, i = 1, .. ., k.
T h e addition of these connector vertices produces k new
t h e gadget for k = 4 is given in Figure 3.22.
C;s.
A n example of
C1H.4PTER3. DIRECTED GRAPH PACKIIVG'S
Figure 3.22: Gadget for t h e {c4)-factor
problem
As in o u r o t h e r proofs, we construct a g r a p h G with vertex set Wl UM/rzU- -UWk.
For each k-tuple of T = ( ( t l , ... , t k ) l t l E CVI,. . .,tk E M/i,} we a d d a copy of
the gadget t o G by identifying connector vertex a1 with vertex t 1 , a;!with t 2 e t
cetera.
Now, we claim t h a t 7 has a k-dimensional matching if and only if G has a {ek)factor. O n c e again, consider t h e case k = 4. If T h a s a +dimensional matching,
then for a 4-tuple in t h e matching, the corresponding copy of t h e gadget in G
has the ( G I - f a c t o r of Figure 3.23.
If a C t u p l e is not in the matching then we have t h e {c4}-factor of Figure 3.24
of t h a t copy of t h e gadget in G.
Thus G has a {GI-factor.
We continue t h e argument as in the proof of Theorem 3.4.1 t o prove t h a t kDM
cc { G I - f a c t o r problem and so the {Ck)-factor problem is NP-complete.
3.5
Packing with Directed Stars
The 15;)-factor problem is polynomial as shown in Theorem 3.2.1.
T h e {$}-factor problem is NP-complete. T h i s can b e proven by reduction of 3DM.
T h e gadget used is a modification of the gadget for t h e ( 6 ) - f a c t o r problem a n d is shown
in Figure 3.25. T h e proof of NP-completeness is similar to t h a t of Theorem 2.3.1.
C'H.4 PTER 3. DIR EC'TED G RA PI1 PAC~l<ltVGS
F i g u r e 3.23: A {c4}-factor containing t h e connector vertices
W e will give a full proof of t h e NP-completeness of t h e { G I - f a c t o r problem a n d then
show how t o c o n s t r u c t a gadget which c a n b e used t o prove t h e .1VP-completeness of any
{,$;.-factor problem
through reduction of the k
+ 1-dimensional
matching problem. T h e
NP-completeness o f t h e {.S%}-factor problem will not b e shown as it is similar t o t h a t of
t h e {c3}-factor problem with t h e obvious alterations a n d renaming.
The Family
G = S3
Theorem 3.5.1. The {GI-factor problem is NP-complete.
This problem is in the class JVP.We will reduce 4DM to t h e {&}-factor problem
t o prove NP-completeness. T h e gadget for this problem is given in Figure 3.26.
We will c o n s t r u c t t h e g r a p h G o n vertex set
( t t , t 2 ,t3, t 4 ) of
W lu W2uW3UW4. For each &tuple
T , a d d a copy of t h e gadget to G by identifying a1 with t l , an
with t 2 , ag w i t h t 3 , a n d
a4
Claim 3.5.1. 7 has
4-dimensional matching if and only if G has an
o
with t 4 .
{GI-
factor.
Proof: S u p p o s e there is a 4-dimensional matching of 7. T h e n for
4-tuples in t h e matching, we have t h e {&}-factor of Figure 3.27 for
t h e c o p y of gadget associated with t h a t 4-tuple in G.
C'H.4 P T E R 3. DIRECTED G R-\ PN P.4CI\'INC;S
Figure 3.21: A {c4}-factor excluding t h e connector vertices
Figure 3.25: Gadget for t h e {&}-factor problem
For those 4-tuples which a r e not in t h e matching, we have t h e factor
of Figure 3.28 of t h e corresponding copy of t h e gadget.
Suppose G has an {&}-factor.
Consider a copy of of t h e gadget. If
a1 is covered by t h e
6 centred
order to cover y, t h e
& with vertices a,a,1 2 a n d ZJ must
{$}-factor.
at
t l
a n d n o t containing y, then in
be in t h e
This would produce t h e factor shown in Figure 3.27. In
this case, we choose t h e 4-tuple ( t l , t s , tS, t4) to be in t h e 4-dimensional
matching of T .
If a, is n o t covered by t h e
& centred
at X I a n d not containing y, then
a similar argument proves t h a t t h e factor of Figure 3.27 is t h e only
one possible a n d we d o not choose t h e $-tuple ( t l , t 2 , t 3 , l 4 ) t o be in
CH.4 P T E R 3. DIRECTED GR.4 P H P.4CKIfNC;S
adget for t h e {%)-lacto
Figu r
the 4-dimensional matching of T.
This gives a 4-dimensional matching of T . 0
Therefore, the {$}-factor
problem is JW-complete. O
The Family 4 = $
A s for t h e {ck)-factor problem, we will show how t o construct a gadget for any k t o be
used in proving the JW-completeness of the {s~J-factor problem.
Once again, we will
demonstrate this construction for t h e case k = 4.
Theorem 3.5.2. The
{.$;I -factor
Construction: Consider a n
problem is NP-complete.
5 centred
at a vertex y and having arcs ( y , zi),
+ 1 Attach
a n & centred a t a vertex zl t o y with the arc zly.
Similarly,attach a n Sk centred at a vertex xi, i = ( 2 , . . . , k + 1) t o each vertex
i=2 . .k
with the arc ziz;. Choose one of t h e unnamed vertices of each star centred at
xi and name it a;,i = 1, ...,k + 1. These are the k + 1 connector vertices of t h e
z;
gadget. When k = 4, this construction produces the gadget of Figure 3.29.
To prove Theorem 3.5.2, reduce the k
+ 1-dimensional matching problem t o t h e
{&}-factor problem as before. The rest of t h e proof is left t o t h e reader. 0
CHAPTER 3. DIRECTED GRAPH R4CKINGS
Figure 3-27: An {&}-factor containing the connector vertices
It"
Figure 3.28: An
{s)-factor excluding the connector vertices
CHA P T E R .3. DIRECTED GRAPH P.4CKlNGS
a4
a3
Figure 3.29: Gadget for the (.!&}-factor problem
Chapter 4
Conclusion
The purpose of this thesis was t o s t u d y t h e g r a p h packing problem for directed graphs.
Much work had been done o n t h e undirected g r a p h packing problem and so we felt t h a t
s t u d y of t h e directed cases might yield interesting results, O u r work was based upon both
matching theory a n d t h e work of many mathematicians on various undirected graph packing
problems. As a result, the second chapter was dedicated to giving t h e reader a n introduction
t o s o m e relevant resuIts on t h e undirected packing problem. In addition, we endeavoured t o
demystify t o some extent certain a r e a s of JVP-completeness theory pertinent t o o u r work.
W e t r u s t t h a t after reading this thesis, t h e reader h a s some understanding of the realm of
NP-completeness a n d of t h e usefulness of this theory in studying and proving the various
degrees of difficulty of problems. O u r NP-completeness proof technique of choice is the
local replacement technique which employs t h e use of gadgets as demonstrated extensively
in C h a p t e r 3. O t h e r methods of proof exist and t h e reader is encouraged t o peruse [24].
We d o n o t claim t h a t C h a p t e r 2 provides a complete survey of all work done o n undirected g r a p h packings. T h e g r a p h packing problem has been approached a n d defined in so
many different ways t h a t a complete survey would produce many volumes of work. We d o
hope however, t h a t Chapter 2 provides a good representation of t h e work done on t h e type
of undirected g r a p h packing problems which we chose t o study in the directed case.
In C h a p t e r 3, we studied directed g r a p h packing problems for families of directed paths,
directed cycles a n d directed stars. T h e focus of t h e chapter was the statement a n d proof
of t h e main theorem of t h e thesis, which s t a t e s t h a t t h e directed G-packing problem is
polynomial for t h e family
G=
{e,6).
To conclude, we provide a s u m m a r y of t h e complexity of t h e problems presented in this
tliesis. Recall t h a t a polynomial problem can be solvecl by a polynomial-time algorithm.
\vhether such a n algorithm has been found or not is another question! JVP-completeness is
a classification of problems of a certain degree of difficulty. Why is it interesting t o know
t h a t a problem is one of these difficult types? Well, t o answer t h a t , we defer t o a nicely
stated explanation from ['24].
Indeed, discovering t h a t a problem is N P - c o m p l e t e is usually just t h e beginning
of work on t h e problem.
. .. However, t h e knowledge t h a t i t is JVP-complete does
provide valuable information a b o u t what lines of approach have t h e potential
of being most productive. Certainly t h e search for a n efficient, exact algorithm
should be accorded low priority. It is now more appropriate t o concentrate o n
For example, you might look for efficient
algorithms t h a t solve various special cases of t h e general problem. You might
other, less ambitious, approaches.
look for algorithms t h a t , though not guaranteed t o run quickly. seem likely t o d o
s o most of t h e time.
... In
short, t h e primary application of t h e theory of JVP-
completeness is to assist algorithm designers in directing their problem-solving
efforts toward those approaches t h a t have t h e greatest likelihood of leading t o
usefuI algorithms.
Table 4.1: Complexity of t h e packing problem for families of cliques
Family
{Ir'l)
{ It-2 1
{a '
4G
NP-complete
Reference
0
[20, 19, '221
up
I&)
{ICn}, n 2 3
{ I(-2, K3 )
G
{K1,&, K3, . . .},
€
G o r I<2 E G
G
{Ii-19f<2,K31...}9 K1,
1-2
Polynomial
0
[361
135, 1361
P I
0
[ l i , 27, 29, 361
~ 9 1
CW.4 PTE R 4. CONCL USlON
Table 4.2: Complexity of t h e packing problem for families of complete bipartite g r a p h s
Family
, {sl,
s2. - - - & )
S = {S1,S2,. . . , I
S n o t a sequential s e t of s t a r s
.
Polynomial
NP-complete
Reference
128, 301
['28, 301
[28, 30, 361
Table 4.3: Complexity of t h e packing problem for families of paths
Family
k
'2
>
{Pk),
I
9
7
Polynomial
AfP-Complete
P2)
Reference
'
[3, 281
Table 4.4: Complexity of t h e packing problem for families of cycles
Family
{C;,C4,. ..) (Two-factors)
{C3?C5,C6?
- -)
{ck,CS, . . .1
{C~,CS~-{C3,C4,.. .1
{ c 3 , c 4 , c 6c~7 - - - )
{C3,C5tC7,- - - )
{CS,
c"lC71- - )
(c6:
c
7
9c81- -)
a)
All o t h e r G
Polynomial
JVP-Complete
Reference
[20, 22, 29, 391
P93
1291
P93
P~I
El8]
B8]
[I8]
R8]
Table 4.S: Complexity of the packing problem for mixed families
I
Familv
hypomatchable
{K2,
H ) , H perfectly rnatch-
I lK2. H I , H
I hypomatchable,
Polynomial
a
perfectly
[ matchable
Table 4.6: Complexity of the packing problem in the directed case
Family
Polynomial
a
{PI)
I JVP-Complete
Reference
{ A ) ,k 2 2
- {PI, ~
a
a
2 )
{PI,q),for j odd, j > 3
{PI,Pi},
for j even, j 2 4
Bc
{ A , . . ., PLL..
.}, 26-2
1 > k , k 2 2 7 f i + ~@ G
{ck), k
3
{Sk),k L 2
4
>
[I21
a
2
a
a
a
CW.4 PTER 4. CONCL USION
Much remains t o be studied in the area of directed graph packings. For example. packing
problems with families of bipartite or complete bipartite directed graphs, families of two or
more directed cycles or directed stars have not been considered. We hope that the work in
this thesis provides a helpful beginning t o the study of the directed graph packing problem.
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