Branch Decompositions and Their Applications

RICE UNIVERSITY

Branch Decomp ositions and their Applications

Illya V Hicks

A Thesis Submitted

in Partial Fulfillment of the

Requirements for the Degree

Do ctor of Philosophy

Approved Thesis Committee

William J Co ok Chairman

Noah G Harding Professor of

Computational and Applied Mathematics

Nathaniel Dean

Asso ciate Professor of Computational and

Applied Mathematics

Alan Cox

Asso ciate Professor of Computer Science

Da vid Applegate

Asso ciate Professor of Computational and

Applied Mathematics

Houston Texas

April

Abstract

Branch Decomp ositions and their Applications

Illya V Hicks

Many reallife problems can b e mo deled as optimization or decision problems

on graphs Also many of those reallife problems are NPhard One traditional

metho d to solve these problems is by branch and b ound while another metho d is

by graph decomp ositions In the s Rob ertson and Seymour conceived of two

new ways to decomp ose the graph in order to solve these problems These ingenious

ideas were only byproducts of their work proving Wagners Conjecture A branch

decomp osition is one of these ideas A pap er by Arnborg Lagergren and Seese showed

that many NPcomplete problems can b e solved in p olynomial time using divide

and conquer techniques on input graphs with b ounded branchwidth but a pap er by

Seymour and Thomas proved that computing an optimal branch decomp osition is also

NPcomplete Although computing optimal branch decomp ositions is NPcomplete

there is a plethora of theory ab out branchwidth and branch decomp ositions For

example a pap er by Seymour and Thomas oered a p olynomial time algorithm to

compute the branchwidth and optimal branch decomp osition for planar graphs This

do ctoral research is concentrated on constructing branch decomp ositions for graphs

and using branch decomp ositions to solve NPcomplete problems mo deled on graphs

In particular a heuristic to compute nearoptimal branch decomp ositions is presented

iii

and the heuristic is compared to previous heuristics in the sub ject Furthermore a

practical implementation of an algorithm given in a pap er by Seymour and Thomas for

computing optimal branch decomp ositions of planar graphs is implemented with the

addition of heuristics to give the algorithm a divide and conquer design In addition

this work includes a theoretical result relating the branchwidth of planar graphs to

their duals characterizations of branchwidth for Halin and chordal graphs Also this

work presents an algorithm for minor containment using a branch decomp osition and

a parallel implementation of the heuristic for general graphs using pthreads

Acknowledgments

First and foremost I would like to thank Go d for blessing me to b e in this situation

He has b een there with me from Waco through SWT and to Rice and I pray that he

will continue to bless and strengthen me SecondlyIwould liketo thank mywife

Casmin for her patience and endurance I would also like to thank my parents Mr

and Mrs Lewis and Lorine Hicks my brother Sedrick Hicks and his family and the

rest of my extended family and friends for their continued supp ort Next I would like

to thank Dr Co ok and Dr Tapia for mentoring me through these graduate years

Your advice and consultation have b een valuable FinallyI would like to thank my

committee members and the rest of the departmentthat gave me a family atmosphere

for these past years

Iwould also like to take the time to acknowledge my grandparents who have passed

away b efore they could see me earn a do ctorate degree I would like to dedicate this

thesis to them b ecause I stand on their shoulders and hop efully my descendants will

stand on mine So I dedicate this thesis to Mr and Mrs Willie and Isab ella Hicks

and Rev and Mrs Andrew and Gilb erta Go o den Other p eople that have passed

awaythatIwould also liketo acknowledge are Mrs Dorothy Collins Mrs Johnetta

Willie Mrs Ruth Evans Mrs Shirley Eldridge and Mrs Lonnie B Ho dges

Contents

Abstract ii

Acknowledgments iv

List of Illustrations vii

List of Tables ix

Introduction

Branchwidth Treewidth and other Denitions

Branchwidth

Branchwidth and The Petersen Graph

Treewidth

Finding Tree Decomp ositions

The Graph Minors Pro ject

Motivation

Finding Branch Decomp ositions

Tree Building

Eigenvector Metho d

Diameter Metho d

Test Instances

Computational Results

Planar Branch Decomp ositions

Ratcatcher Metho d

Carvingwidth

Antipo dality

The Algorithm

CycleMethod

Edge Contraction Metho d

Computational Results

Planar Graphs and Their Duals

Characterizing Branchwidth

Halin Branchwidth

Chordal Branchwidth

Branch Decomp osition Applications

Minimum Fillin

Minor Containment

Edge Discrimination

Maximum Clique

Computational Results

Parallel Branch Decomp ositions

Conclusions and Future Work

Bibliography

Illustrations

Example graph

Optimal branch decomp osition of width for Figure

Obstruction set for graphs with branchwidth at most

Tangle of order for Figure

The Petersen graph and an optimal branch decomp osition

Tangle of order for the Petersen graph

Maximal prop er minors of the Petersen graph

Optimal branch decomp osition for pm

Optimal tree decomp osition for Figure

Obstruction set for graphs with treewidth at most

Optimal branch decomp osition for Figure

The initial split

Subsequent splits if jX j

Example G

Example G and G

x y

Planar Test Instances with their branchwidth

Compiler Test Instances with their branchwidth

Fillin Test Instances with their branchwidth

viii

Q from Chapter

Medial graph of Figure

Example hypergraph H

Medial graph of Figure

Corresp onding dual graph of Figure

Resulting hypergraphs from using path P

Some Test Instances with their corresp onding branchwidth

M has branchwidth

The branchwidthofHalingraphsis

The minor set for a leaf

Joining minor set graphs for a nonleaf no de p

Edge discrimination

The maximum clique in the conict graph of this minor set graph is

Do es tele contain the Petersen graph P as a minor

The toroidal grid C x C

rkfs has branchwidth

Tables

Planar graphs

Some more planar graphs

Compiler graphs

More compiler graphs

Minimum Fillin graphs

Cycle Metho d versus Edge Contraction Metho d

Cycle Metho d Results

Classes of Graphs and their branchwidth

Minimum Fillin Results

Minor Containment Results

Parallel implementation for planar graphs

Parallel implementation for more planar graphs

Chapter

Intro duction

Many reallife problems can b e mo deled as optimization or decision problems on

graphs Consider for instance the problem in whicha spy has to visit a number of

cities and return to home base The spy can not revisit a city in fear of capture and

b ecause of budget cuts the spy has to follow the shortest route visiting all of the

sp ecied cities and returning to home base The problem is formally known as the

traveling salesman problem The input can b e mo deled as a weighted graph in which

the no des represent the dierent cities and home base and an edge b etween twonodes

represents the path b etween the corresp onding cities and has weight that corresp onds

to the cost of the spytraveling on that particular route The problem is to nd a

circuit in the graph whichcontains all no des and has minimum total weight

Unfortunately many graph problems like the one mentioned ab ove are hard in

the sense that there are probably no ecient algorithms to solve them One wayto

overcome this disadvantage is by decomp osing the graph into subgraphs such that

the structure of the input graph would assure that the problem on the subgraphs are

easy to solve Such a decomp osition structure might help in nding a more ecient

algorithm that computes an optimal solution or a go o d approximate solution for

the complete problem One suitable structure is the tree structure b ecause many

graph problems that are hard in general are eciently solvable on trees One type of

decomp osition that has a tree structure is the branch decomp osition

Branch decomp ositions were introduced in Rob ertson and Seymour as part of

their Graph Minors Pro ject a series of pap ers containing the pro of of Wagners conjec

ture Branch decomp ositions op en algorithmic p ossibilities for intractable problems

that can b e mo deled on graphs Arnborg Lagergren and Seese showed that many

NPcomplete problems can b e solved in p olynomial time using divide and conquer

techniques on input graphs with b ounded branchwidth but Seymour and Thomas

proved that computing an optimal branch decomp osition is also NPcomplete In this

do ctoral work the problem of how to compute nearoptimal and optimal branchde

comp ositions is addressed branch decomp ositions were also used to solve problems

like minor containment and minimum llin In Seymour and Thomas the au

thors gave an algorithm to compute the branchwidth of any planar graph This

work includes an implementation of that algorithm as a b enchmark for the aforemen

tioned heuristics for branch decomp ositions of general graphs The implementation

of the Seymour and Thomas algorithm includes the addition of heuristics to give

the algorithm a divide and conquer design This work also contains a theoretical

result relating the branchwidth of planar graphs to their duals characterizations of

branchwidth for some sp ecic classes of graphs and a parallel implementation of the

heuristic for general graphs using the POSIX threads library which is also known as

pthreads

Results of this thesis are presented as follows Ma jor denitions and terms are

discussed in Chapter In Chapter some background is given on the Graph

Minors Pro ject and how branch decomp ositions were used in the pro of of Wagners

Conjecture Chapter is a discussion of the many algorithmic p ossibilities of branch

decomp ositions Heuristics for nearoptimal branch decomp ositions are discussed in

Chapter while our implementation of the Seymour and Thomas branchwidth al

gorithm and our improvement of the algorithm for optimal branch decomp ositions

is detailed in Chapter Chapter oers characterizations for the branchwidth of

two classes of graphs In Chapter a heuristic for the minimum llin problem

and an implementation of an algorithm to test for minor containment in graphs with

b ounded branchwidth are given Chapter oers results from a parallel implemen

tation of some of the heuristics for nearoptimal branch decomp ositions Conclusions

and future work are discussed in Chapter

Chapter

Branchwidth Treewidth and other Denitions

It is imp erativethat branch decompositions and branchwidth b e dened and under

sto o d b efore undertaking the explanations of algorithms results and conclusions of

this work Once foundational denitions and terms are introduced they will b e used

in an exercise to oer another characterization of the infamous Petersen graph Other

helpful terms and structures will also b e dened to conclude this chapter b ecause they

have some relationship with branchwidth and are more p opular in the literature on

the sub ject

Branchwidth

Let G b e a graph with no de set V G and edge set E G Let T b e a tree having

jE Gj leaves in whichevery nonleaf no de has degree Let b e a bijection from the

edges of G to the leaves of T The pair T iscalledabranch decomposition of G

Notice that removing an edge say eof T partitions the edges of G into two subsets

A and B The midd le set of e and A B denoted by midA B ormide is

e e e e e e

the set V A V B The width of a branch decomp osition T is the maximum

e e

order of the middle sets over all edges in T Thebranchwidth of G denoted by G

is the minimum width over all branch decomp ositions of G A branch decomp osition

of G is optimal if its width is equal to the branchwidth of G The pair T is called

a partial branch decomposition if T has a no de with degree greater than Figure

gives an optimal branch decomp osition of Figure where some of the middle sets

of the edges of Figure are provided Now that some foundational denitions are

given this section is continued with the examination of some prop erties of branch decomp ositions

021 3

bc 456 7 89h d efg 11 12 10 13 ij

14 15 16 17

Figure Example graph

14 15

2 3 4 11 8

{b, d, f, k} {a, c, h} {a, f, k} 5 6 {a, b, d} {c, f, h, k} 17

16 12 7 0 1

13 9

Figure Optimal branch decomp osition of width for Figure

An hyperedge e is contracted if e is deleted and the ends of e are identied into

one no de A graph H is a minor of a graph G if H can b e obtained from a subgraph

of G bycontracting edges Let F b e a class of graphs F is minor closed when all

the minors of any member of F also b elong to F Theorem given b elow relates

the branchwidth of graphs to the branchwidth of their minors

Theorem Robertson and Seymour If H is a minor of graph

G then H G

A subdivision of a graph G is a graph obtained from G by replacing its edges by

internally vertexdisjoint paths So Corollary stated b elow can b e derived from

Theorem which can b e useful in determining the branchwidth of graphs This

corollary is esp ecially imp ortant for a pro of in Section

Corollary If G is a graph and G andH is a sub division of

G then G H

K5 M6

Figure Obstruction set for graphs with branchwidth at most

Theorem also reveals that given some k the class of graphs with branch

width at most k is a minor closed class of graphs Given a minor closed class of

graphs F theobstruction set of F is the set of minor minimal graphs that are not

elements of F In addition one could prove that a graph G has branchwidth at most

k if one could showthatG contains no minor in the obstruction set of graphs with

branchwidth at most k Rob ertson and Seymour proved that any minor closed

class of graphs contains a nite set of minor minimal elements So any minor closed

class of graphs has a nite obstruction set For example Bo dlaender proved that

the four graphs given in Figure are the only graphs in the obstruction set for the

class of graphs with branchwidth at most The obstruction set for branchwidth at

most k for anyinteger k greater than is not known Graphs of smaller branchwidth

are characterized by the following theorem

Theorem Robertson and Seymour A graph G has branch

width

if and only if every comp onentof G has edge

if and only if every comp onentof G has nodeofdegree

if and only if G has no K minor

So one factors have branchwidth equal to Stars trees with only one nonleaf

no de are the only connected graphs with branchwidth equal to Thus all forests

that are not the union of one factors and stars have branchwidth equal to Also

seriesparallel graphs and outerplanar graphs have branchwidth at most b ecause

they do not contain K as a minor Let n be aninteger Let H b e a simple

graph with V H fi j i j ngwherei j and i j are adjacentif

o o

ji ij jj j j H is called an ngrid Rob ertson and Seymour

o o

proved that the branchwidth of an ngrid is n The following theorem characterizes

the branchwidth of complete graphs

Theorem Robertson and Seymour Let G b e a complete graph

with jV Gj then Gd jV Gje

For example Theorem together with Theorem directs one to conclude that

graphs with K as a minor have branchwidth at least

0 a b

2 4

ef3

1 5 6 7

gh8

10 9

cd11

Figure Q

In order to discuss more prop erties involving branchwidth several more denitions

are needed Let G b e a graph and let k beaninteger A separation of a graph

G is a pair G G of subgraphs of G with G G V G V G EG

E G G and E G E G and the order of this separation is dened as

f jV G V G jAtangle in G of order k is a set T of separations of G eacho

order k such that

T for every separation A B of G of order k one of A B B Ais an

elementof T

T if A B A B A B T then A A A G and

T if A B T then V A V G

Separation of order

Separations of order

v G v V G

Separations of order

fv wg G v w V G

e G n e e E G

Separations of order

fv w ug G v w u V G

fv eg G n e e E Gandv V Gsuch that v is not incidentto e

f E Gsuch that e and f share a no de fe f g G nfe f g pairs e

f g G n a

f g G n b

f g G n c

f g G n e

f g G n f

f g G n d

f g G n g

f g G n h

Figure Tangle of order for Figure

These are called the rst second and third tangle axioms The tangle number of

G denoted by G is the maximum order of any tangle in G Figure gives an

example of a tangle of order for Figure Notice in Figure that the inclusion

of separations of the graph of order to the tangle would result in a violation of

one of the tangle axioms A tangle T of G with order k can b e thought of as a k

connected comp onentof G b ecause some kconnected comp onentof G will either

b e on one side or the other for any separation of T Rob ertson and Seymour

proved a strong minmax relation b etween tangles and branch decomp ositions

Theorem Robertson and Seymour For any simple graph G

such that E G max G G

The remainder of this section is dedicated to oer a pro of of the weak duality of

Theorem as Theorem The two lemmas that follow will assist in the pro of

Lemma Robertson and Seymour If T is a tangle of order k

and A B A B T and A A B B has order k then

A A B B T

Lemma Robertson and Seymour Let T b e a set of separa

tions of graph Geach with order kand k satisfying the rst

and second tangle axioms Then T is a tangle of order k if and only if

e G n e Te E G

Theorem Let G b e a simple graph such that G andE G

T hen G G

Pro of Without loss of generality assume G has no isolated no des

Supp ose G G Let T b e a tangle of order G and let T

b e a branch decomp osition with width G Let v b e a nonleaf no de of

T RootT at v Soevery nonleaf no de has twochildren x and y except

for v which has three children Visit each no de except for v in p ostorder

and let a bijection from the no des of T to ordered separations of Gbe

dened as

uG n u if u is a leaf no de

A B xand A B y A A B B where

Thus u T for every u V G n v b ecause of Lemma and

Lemma So v has three children and the corresp onding separations

are A B C B A C and C A B whereA B and C are edge

disjoint subgraphs of G b ecause T is a branch decomp osition and by

the denition of SoA B C B A C and C A B are

all in T but A B C G which contradicts tangle axiom T

G G

Therefore D

Branchwidth and The P etersen Graph

The Petersen graph has b een an imp ortant part of the development of graph theory

Many conjectures ab out vertex and edge colorability factors ows and other graph

invariants have b een disproved by using the Petersen graph as a counterexample

Here however the Petersen graph will b e used to illustrate the usage of some of the

theory and terms discussed in the previous section This section will prove that the

branchwidth of the Petersen graph is and also provethat the Petersen graph is a

member of the obstruction set for the class of graphs with branchwidth at most

Figure illustrates an optimal branch decomp osition of width for the Petersen

graph

The branch decomp osition given in Figure oers us an upp er b ound on the

branchwidth of the graph In order to proveequalitya lower b ound equal to the

upp er b ound will need to b e established Recall from Theorem in Section

that if the branchwidth of a graph G is greater than then the order of anytangle

of G isavalid lower b ound to the branchwidth of GObviously the Petersen graph

has branchwidth greater than b ecause it has K as a minor A tangle of order for

the Petersen graph is given in Figure Notice in Figure P represents a path

on k no des Also notice that these are all the separations of order strictly less than

and that the union of the left sides of any three separations in the tangle will not

etersen graph Thus wehave the following result b e isomorphic to the P

Theorem The Petersen graph has branchwidth equal to

045 6 9 12 14 11 1 10 3 13 7 8 2 3 8 11 10 613 2 14 12

0 5

1 7 4 9

Figure The Petersen graph and an optimal branch decomp osition

It has b een the goal of this section to provethat the Petersen graph is indeed a

member of the obstruction set of the class of graphs with branchwidth at most In

order to prove this result and achieve the goal of this section one must show that

any prop er minor of the Petersen graph has branchwidth

Theorem The Petersen graph is a member of the obstruction set

of the class of graphs with branchwidth at most

Pro of From Theorem the branchwidth of the graph is Since the

Petersen graph is edge transitive all prop er minors of the Petersen

graph are isomorphic to a minor of one of the two graphs given in Figure

Nowonemust show that pmandpm b oth have branchwidth equal to

Obviously pmcontains K as a minor So pm has branchwidth at

least But one can easily nd a branch decomp osition of width for

pm Such a branch decomp osition for pmisgiven in Figure For

Separation of order

Separations of order

v G v V G

Separations of order

fv wg G v w V G

e G n e e E G

Separations of order

fv w ug G v w u V G

fv eg G n e e E Gandv V Gsuch that v is not incidentto e

G P G n E P P pathonnodes

Gu v w x G n u u v w x V Gsuch that v w x adj u

Separations of order

fv w u xg G v w u x V G

fv u eg G n e e E Gandv u V Gsuchthat v and u are not

incidentto e

v P G n E P P path on no des G and v V G sucht hat v is

not a part of P

P G n E P P pathonnodes G

fe f g G nfe f g pairs e f E Gsuch that e and f do not share a no de

Gu v w x z G n E Gu v w x z u u v w x z V Gsuch that

v w x adj u

Guv adj uadj v G nfu v g uv E G

Figure T angle of order for the Petersen graph

pm one can observethat pm is a sub division of M which is illustrated

in Figure So by Corollary the branchwidth of pmis Thus the

Petersen graph is a member of the obstruction set of the class of graphs

branchwidth at most

with D

Treewidth

ertson and Seymour the authors conceived a new way to decomp ose a In Rob

graph to display the similarities b etween the graph and trees This concept tree

pm1 pm2

Figure Maximal prop er minors of the Petersen graph

0 4 5 9 13 8 11

1 3 10 12 6 7 2

1 2 511 0 9

6 12 4 13

73 810

Figure Optimal branch decomp osition for pm

decompositions and the parameter asso ciated with the concept treewidthwere in

tro duced b efore branch decomp ositions and branchwidth Branchwidth is the brother

of treewidth which has b een extensively researched by Thomas Seymour and

Thomas Bo dlaender and Kloks Bo dlaender Ramachandramurthi

Reed and many others see the survey pap ers by Bo dlaender The

dierence b etween branchwidth and treewidth is that branchwidth deals with edges

and treewidth deals with no des

Let G b e a graph and let T b e a tree For v V T v is a subgraph of G

The pair T is called a treedecomposition of G if

v v V T G

for distinct v w V T Ev E v

for all uv w V T if v is on the path from u to w in T thenu w v

V v j for all v V T The width of a tree decomp osition is the maximum of j

The treewidth of G denoted by G is the minimum width over all tree decomp osi

tions of G G has treewidth at most k if there is some tree decomp osition with width

equal to k A tree decomp osition of G is optimal if its width is equal to the treewidth

of GIfT is restricted to b e a path then T is called a path decomposition and the

pathwidth of the graph G denoted by G is the minimum width for any path de

comp osition of GPathwidth was intro duced by Rob ertson and Seymour where

the authors explored the algorithmic capabilities of treewidth Furthermore one can

see that the pathwidth of a graph G is at least the treewidth of G

23 19 04 78 5 6 11 10

width = 3

Figure Optimal tree decomp osition for Figure

Figure gives an optimal tree decomp osition of Figure In Figure for

anodev of the tree v is the the induced subgraph of the no des or edges lab eled on

v The graphs of small treewidth are characterized by the following theorem stated

in Bo dlaender

Theorem

A graph G has treewidth at most if and only if G do es not contain

K as a minor

A graph G has treewidth at most if and only if G do es not contain

K as a minor

A direct corollary of Theorem and Theorem is the fact that a graph has

branchwidth at most if and only if the graph has treewidth at most In addition

Figure is an optimal tree decomp osition of Figure b ecause K is a minor

of Figure Treewidth and tree decomp ositions have b een investigated more than

branchwidth Some minmax relationships for treewidth are discussed by Seymour and

Thomas and by Reed The treewidth of some wellstudied graph classes are

also known Bo dlaender revealed in the authors survey pap er on treewidth that

forests have treewidth equal to seriesparallel have treewidth equal to outerplanar

graphs have treewidth at most Halin graphs have treewidth equal to and complete

graphs have treewidth equal to the order the graph minus For an ngrid H

Rob ertson and Seymour proved that the treewidth of H is at least n There

is also a relationship b etween graphs and their minors for treewidth and pathwidth

Theorem Robertson and Seymour If H is a minor of graph

G then H G and H G

Theorem states that the class of graphs with treewidth at most k and the class

of graphs with pathwidth at most k are also minor closed classes So graphs with

K5 M6

M10

Figure Obstruction set for graphs with treewidth at most

treewidth at most k and pathwidth at most k also have nite obstruction sets So one

could prove that graph G has treewidth at most k if one shows that G has no minor in

the obstruction set of graphs with treewidth at most k and resp ectively for pathwidth

Arnborg Proskurowski and Corneil and Satyanarayana and Tung proved that

the graphs in Figure are the only graphs in the obstruction set for graphs with

treewidth at most Ramachandramurthi gavelower b ounds for the number of

graphs in the obstruction set of graphs with treewidth at most through and

The obstruction sets of graph with pathwidth and are also known The

size of the obstruction sets can grow rapidly the obstruction set of the graphs with

pathwidth at most k contains at least k graphs eachcontaining O no des

In Rob ertson and Seymour the following relationship b etween branchwidth

and treewidth is proven

Theorem Robertson and Seymour For any simple graph G

such that E G max G G maxb Gc

So one could nd tree decomp ositions of a graph by nding a branch decomp osi

tion of the graph and vice versa Figure is an example of an optimal branch

decomp osition of Figure Figure and K are examples where b oth extremes

of Theorem can o ccur

2 3 78

6 10 {a, f, g} {d, f, g}

{a, d, f, g}

{a, f, d} {a, d, g}

5 11

0419

width = 4

Figure Optimal branch decomp osition for Figure

Agraph G is a ktree for some integer k if and only if either G is isomorphic to

K or there exists a k tree H that is a subgraph of G such that G can b e constructed

from H and a no de v that is adjacentto ak clique in H Apartial k tree is a subgraph

of a k tree By construction k trees have a maximum clique number of k and

have treewidth equal to k Below is a theorem relating partial k trees with treewidth

Theorem van Leeuwen A graph G is a partial k tree if and

only if the treewidth of G is at most k

Thus if one could prove that a particular graph is a partial k tree for some k then

that graph has treewidth at most k In addition for a particular graph G if one

found the smallest k such that G is a subgraph of some k tree then k would b e the

treewidth of G

Finding Tree Decomp ositions

Since this thesis is dedicated to the construction and usage of branch decomp osi

tions instead of tree decomp ositions and tree decomp ositions are more p opular than

branch decomp ositions in the literature the nal section of this chapter summarizes

some algorithms to construct tree decomp ositions for graphs Arnb org Corneil and

Proskurowski stated that determining whether the treewidth or pathwidth of a

given graph is at most a given integer k is NPcomplete The complexity of these

problems has also b een studied for several classes of graphs

For approximate algorithms Bo dlaender Gilb ert Hafsteinsson and Kloks

gave a p olynomial time approximation algorithm with O log n p erformance ratio

for treewidth and a p olynomial time approximation algorithm with O log n p erfor

mance ratio for pathwidth Also for several classes of p erfect graphs Bo dlaender

and Kloks oer p olynomial time approximation algorithms In addition using

Theorem and the p olynomial time algorithm by Seymour and Thomas for

the branchwidth of planar graphs one can nd a p olynomial time approximate tree

decomp osition for any planar graph

Bo dlaender describ ed an algorithm to determine the treewidth of a graph in

linear time with resp ect to the size of the input Although this algorithm app ears to

b e impractical practical algorithms have b een presented for graphs with treewidth

and and Arnborg Corneil and Proskurowski proved that

nding the treewidth of a graph is solvable in O n t imewherek is a constant

Lagergren gave a sequential algorithm that uses O nl og n time and a parallel

variant that uses O n pro cessors and O log n time Both algorithms either deter

mine that the input graph G has treewidth more than some constant k or nd a

treedecomp osition of G whose width is b ounded by some factor of k Reed im

proved Lagergrens sequential algorithm to achieve a complexity O nl og n Matousek

and Thomas gave a probabilistic result with running time O nl og n njlog pj

where p is the error of probability This algorithm nds optimal tree decomp ositions

for graphs with treewidth and Many of these algorithms either determine that the

input graph G has treewidth more than some constant k or nd a treedecomp osition

of G whose width is b ounded by some factor of k Others nd an optimal tree decom

p osition by using an existing tree decomp osition and there are still other algorithms

that work on a minor of the graph in order to determine the treewidth Since

most of the foundational terms used in this work are now dened the following two

chapters are fo cused on the background and motivation of branch decomp ositions resp ectively

Chapter

The Graph Minors Pro ject

Recall from Chapter that branch decomp ositions were introduced by Rob ertson and

Seymour as part of a series of pap ers the Graph Minors Pro ject that combine

to proveWagners Conjecture The purp ose of this chapter is to dene and provide

some background on Wagners Conjecture and to present some information on how

branch decomp ositions playedapartintheproof

A planar graph is a graph that can b e drawn on a sphere or plane without having

edges that cross Recall that a sub division of a graph G is a graph obtained from G by

replacing its edges byinternally vertexdisjoint paths In the s Kuratowski

proved that a graph G is planar if and only if G do es not contain a sub division of

K or K Wagner later proved that a graph G is planar if and only if G do es

not contain K or K as a minor of G The question for other surfaces remained

op en until when Archdeacon and Glover Hunekeand Wang solved the

case for the pro jective plane where they proved that there are minorminimal

nonpro jectiveplanar graphs In the s Erdos p osed the question of whether

the list of minor minimal graphs that cannot b e embedded in a given surface is

nite Archdeacon and Huneke proved that the list for any nonorientable surface

is nite and Rob ertson and Seymour proved the case for any surface with the

Graph Minors Pro ject

Given some surface an antichain is a list of minor minimal graphs which cannot

be emb edded in This means that no member of an antichain is isomorphic to a

minor of another In the early s Wagner conjectured that every antichain is

nite The pro of of this conjecture would imply that the obstruction set for any minor

closed class is nite

Conjecture K Wagner For every innite sequence of graphs G G

there exists i j with ij suchthatG is isomorphic to a minor of G

i j

In order to give more background of Wagners Conjecture a few denitions are

needed A class with a reexive and transitiverelation is a quasiorderFor example

the relation H is isomorphic to a subgraph of G denes a quasiorder on the class

of all graphs A quasiorder denoted byQ is wel lquasiordered if for every

countable sequence q q of members of Q there exist ij suchthat q q

i j

Wagners conjecture is equivalent to stating that the minor quasiorder H is

isomorphic to a minor of G is wellquasiordered

One quasiorder that is not wellquasiordered is the subgraph quasiorder stated

earlier This is true b ecause a countable set of circuit graphs one of each size is an

innite antichain A graph G topological ly contains a graph H if G has a subgraph

which is isomorphic to a sub division of H Top ological containment is not a wellquasi

ordering as well The set of graphs formed by taking a circuit graph of each size and

replacing eachedgebytwo parallel edges is an innite antichain however Kruskal

proved that the class of all trees is wellquasiordered under top ological containment

one of two famous conjectures of Vazsonyi Rob ertson and Seymour used

this theorem to prove

Theorem For anyinteger k the class of all graphs with treewidth

k is wellquasiordered by minors

The other conjecture of Vazsonyi was that the class of all graphs with maximum

degree at most three is a wellquasiordering under top ological containment

Another quasiorder is immersion A pair of adjacentedges ab and bc is lifted if

ab and bc are replaced by the edge ac A graph H is immersed in a graph G if H

can b e obtained from a subgraph of G by lifting pairs of edges NashWilliams

conjectured that the immersion quasiorder is wellquasiordered This would imply

b oth of Vazsonyis conjectures

Now that sucient background of Wagners Conjecture has b een given this work

will sketch a pro of of Wagners Conjecture given by Rob ertson and Seymour

Rob ertson and Seymour stated one waytoproveWagners conjecture is to prove

that for every graph H and every innite set of graphs that do not contain a minor

isomorphic to H some member of the set is isomorphic to another member To this

end supp ose H is planar Rob ertson and Seymour showed that there exists some

number N whichwas later improved by Rob ertson Seymour and Thomas such

that the class of graphs that do not contain a minor isomorphic to H have treewidth

at most N This result was used by Rob ertson and Seymour to show that the

class of graphs that do not contain a minor isomorphic to H is wellquasiordered by

minors a corollary of Theorem So supp ose H is nonplanar For the purp ose

of the rest of the pro of think of a tangle of a graph G as some highly connected

comp onent that lies on one side of every separation of the tangle in the formal

sense It was shown by Rob ertson and Seymour

Theorem For every graph H ifG has no minor isomorphic to H

then every large order tangle of G can almost b e embedded on a surface

on which H cannot b e drawn

The word almost refers to complex denitions of vertex extensions and vortices

that are omitted in this sketch pro of but can b e found in full detail in Rob ertson

and Seymour Since there are only nitely many surfaces where H can not b e

embedded on Rob ertson and Seymour used Theorem to state that in order

to proveWagners conjecture it suces to show that

Theorem If are surfaces then for every innite set F of

graphs if every large order tangle of every member of F can almost

be drawn in one of thensomemember of F is isomorphic to a

minor of another member of F

To prove Theorem Rob ertson and Seymour utilized the main results of Rob ertson

and Seymour and Rob ertson and Seymour The main result of Rob ertson

and Seymour states that if F is an innite set of graphs and all the large order

tangles of anymember G of F are wellbehaved then there exist H and H both

members of F such that H has a minor isomorphic to H The formal denition

of a tangle b eing wellbehaved is omitted in this sketch pro of but can b e found in

Rob ertson and Seymour Toshowthatthehyp othesis of Theorem implies

that all of these large order tangles are wellbehaved Rob ertson and Seymour

used the main result of Rob ertson and Seymour which asserts that for any innite

set of graphs all drawable in a xed surface some member of the set is isomorphic to

a minor of another

Tree decomp ositions were introduced b ecause large order tangles have a treelike

structure in their asso ciation with small order tangles This phenomenon can b e

seen in Figure where the the nonleaf no de is a connected comp onent of

Figure and the leaf no des represent tangles of order Branch decomp ositions

were introduced in Rob ertson and Seymour where they studied the relationship

between tree decomp ositions and tangles

Chapter

Motivation

Why are branch decomp ositions imp ortant Since the edges of branch decomp osi

tions dictate two disjoint set of edges of the graph the edges of branch decomp ositions

allow the graph to b e broken into smaller pieces This characteristic of branchde

comp ositions makes dicult problems on graphs with a given branch decomp osition

more tractable The edges of branch decomp ositions also showhow to rejoin the

pieces by the middle sets Many researchers have oered algorithms that use branch

decomp ositions or tree decomp ositions mostly tree decomp ositions for use on prob

lems like ring routing traveling salesman problem TSP coloring minimum llin

and disjoint paths

In Co ok and Seymour the authors used branch decomp ositions to solve ring

routing problems arising in the design of reliable cost eective SONET networks The

ring routing problem is the following given a network with a cost assigned to each

edge nd a minimum cost circuit passing through eachmemb er of a sp ecied subset

of no des Clearly the traveling salesman problem is a sp ecial case of the ring routing

problem Co ok and Seymour also used branch decomp ositions to nd solutions

to TSP problems One use of branch decomp osition that the authors implemented is

to take the union of two or more solutions of a TSP problem to form a sparse graph

and then use the branch decomp osition of the sparse graph to nd a solution in the

sparse graph that was b etter than the initial solutions This metho dology was used

to pro duce the b est known solutions for the unsolved problems in the TSPLIB

library of standard test instances for the traveling salesman problem

Furthermore certain optimization problems arising in compiling computer pro

grams can b e attacked using branch decomp ositions The register allo cation problem

for an imp erative program P is usually mo deled as the coloring problem of the in

terference graph I of the controlowgraph G of P The interference graph of a

controlow graph G is the intersection graph of some connected subgraphs of G

The controlow graph is a digraph representing the owofcontrol b etween p oints in

the execution of the program P The subgraphs represent the life span of variables

of a program The coloring problem mo dels whether twovariables with overlapping

life spans should b e assigned to dierent registers Thorup showed that if

acontrolow graph has treewidth k then one can eciently color anyintersection

graph of connected subgraphs within a factor of bkc of optimality Since

treewidth and branchwidth are closely related one could use branch decomp ositions

to nd a similar factor

Branchwidth may also playapart ininterval routing schemes Compact routing

metho ds are metho ds where pro cessors decide over what link to forward messages

that take little space for storing such routing information This would b e

considered for a distributed pro cessor network with message passing One type of a

compact routing metho d is interval routing For a k interval routing each pro cessor

is numb ered with a unique integer and each outgoing link is lab eled with at most k

cyclic intervals of integers corresp onding to the pro cessor names A message when

not arriving at its nal destination is forwarded over the link whose lab el has an

interval that contains the name of the destination pro cessor The ob jectiveis to

have the messages arrive at their destinations using the metho d by the shortest

route Bo dlaender Tan Thilikos and van Leeuwen worked on this problem

using treewidth and tree decomp ositions Branchwidth could also b e used for this application

Researchers in molecular biology are interested in a problem that can b e mo deled

on a graph Given a class of sp ecies a set of characteristics and for each sp ecies type

and eachcharacteristic a value that the characteristic has for that sp ecies type the

problem is to nd a go o d evolution tree for these sp ecies and their p ossible extinct

ancestors This problem is called the perfect phylogeny problem This problem is

equivalent to the following graph problem Given a graph G with an existing coloring

of the graph can one add edges to G such that the resulting graph is chordal and the

input coloring is still a valid coloring Bo dlaender Fellows and Kloks tackled

this problem using a tree decomp osition This problem seems to b e more suitable for

branch decomp ositions b ecause of the emphasis on edges

Another p ossible application for branch decomp ositions is VLSI design Bo dlaender

describ ed a well studied problem in VLSI layout theory called the gate matrix layout

problem The problem is p osed in terms of a matrix M whose columns represent

gates G G and whose rows represent nets N N If there is a nonzero entry in

n m

row i and column j then N must b e connected to gate G The ob jectiveis to nd a

i j

p ermutation of the gates such that all nets can b e made within the minimum number

of tracks to lay the nets suchthatnoneoverlap on the same track Bo dlaender also

stated that this problem is equivalent to nding the pathwidth of a graph

This problem and other graph problems related to VLSI design may b e solvable using

a branch decomp osition

In Rob ertson and Seymour the authors gave an impractical algorithm to com

pute disjoint paths using branch decomp ositions Scheer gave a more practical

algorithm for disjoint paths where the author used tree decomp ositions to compute

disjoint paths One can see that Scheers algorithm could b e mo died to actually

compute disjoint paths using branch decomp ositions

Other p ossible applications are in exp ert systems and natural language pro cess

ing Bo dlaender states that graphs mo deling certain types of exp ert systems

have b een observed to have small treewidth in practice Branch decomp ositions could

b e used to p erform statistical computations for reasoning with uncertainty in these

systems For natural language pro cessing Kornai and Tuza observed

that dep endency graphs of sentences enco ding the ma jor syntactic relations among

the words usually have pathwidth at most and that the pathwidth closely resembles

a parameter asso ciated with the graph Branchwidth could also assist in solving this

problem as well

In much of this thesis pap ers dealing with treewidth have b een referred rather

than branchwidth The asp ects of treewidth have b een extensively researched while

branchwidth has b een somewhat ignored For this reason this do ctoral work fo cuses

on branchwidth rather than treewidth The intent of this research is to spark an in

terest in the algorithmic and theoretical asp ects of branchwidth Some NPcomplete

problems seem more suitable for branchwidth rather than treewidth For example

branchwidth seems more suitable to compute optimal or approximate solutions to the

disjoint paths problem Even though a strong heuristic using branch decomp osition

for the minimum llin problem is presented in Chapter branchwidth would b e

suitable to optimally solve the minimum llin problem as well There are several

classes of graphs where the branchwidth of the class is an op en problem Although

one member of the obstruction set for the class of graphs with branchwidthatmost

isprovided in Section the rest of the set is unknown There are several other

obstruction sets that have not b een dened for branchwidth and ab ove Another

reason to fo cus on branchwidth is that there exist more realistic results ab out branch

width than treewidth For example one can compute the branchwidth of any planar

graph in p olynomial time an op en problem for treewidth

Chapter

Finding Branch Decomp ositions

How do es one nd nearoptimal branch decomp ositions In Rob ertson and Seymour

there is a fast algorithm to estimate the branchwidth of a graph within a factor of

For example the algorithm decides if a graph has branchwidth at least or nds a

branch decomp osition with width at most This algorithm has not b een used in

a practical implementation and its improvements by Bo dlaender Bo dlaender and

Kloks and Reed have not b een shown to b e practical either Bo dlaender and

Thilikos did give an algorithm to compute the optimal branch decomp osition for

anychordal graph with maximum clique size at most but the algorithm has b een

only shown practical for a particular type of tree Bo dlaender and Thilikos also

oered a linear time algorithm for nding an optimal branch decomp osition using a

branch decomp osition but that algorithm also app ears to b e impractical Seymour

and Thomas did give a practical algorithm to compute the branchwidth of planar

graphs but for general graphs one has to rely on heuristics Co ok and Seymour

gave a heuristic algorithm to pro duce branch decomp ositions that shows promise

The diameter method and the hybrid method are closely related to the algorithm

All of the heuristics consist of a tree building comp onent and a separation nding

comp onent First tree building will b e discussed Next the chapter fo cuses on the

Co ok and Seymour heuristic the eigenvector metho d followed by the details of the

diameter and hybrid metho ds and computational results comparing the heuristics

Tree Building

Since computing an optimal branch decomp osition is NPcomplete one relies on

heuristic metho ds to obtain nearoptimal branch decomp ositions Let G be a bi

connected graph b ecause one can nd a branch decomp osition of any graph by rst

nding branch decomp ositions of the graphs biconnected comp onents First start

with a partial branch decomp osition T where T is a star with jE Gj leaves

such that each leaf is asso ciated with a distinct edge of G Let v be the node of

T that is adjacent to all other no des For the initial split a separation A B is

found suchthat jE Aj jE B j and a new partial branch decomp osition T

is created where v is replaced bynodes x and y and edge x y and x has the leaves

corresp onding to E Aand y has the leaves corresp onding to E B An initial split

is illustrated in Figure In T let a be a node of T with degree greater than

Let G b e the subgraph of G asso ciated with the neighbors of a that are leaves

By the construction of T a has only one neighbor that is not a leaf let e be the

edge b etween a and this neighbor Lab el the no des of midein G as b eing linking

no des and nd a separation X Y ofG Let us assume that the cardinalityof X is

at most the cardinalityof Y IfjX j create T by creating new no des x and

y and edges a xanda y with x having the leaves corresp onding to E X and y

having the leaves corresp onding to E Y The middle sets of the new edges would

mida x V X V Y mide X

mida y V X V Y mide Y

If jX j create T by creating a new no de y and an edge a y with y having the

leaves corresp onding to E Y The middle set of the new edge would b e the same as

Equation A subsequent split when jX j isgiven in Figure while Figure

gives an example of a subsequent split when jX j Notice that the nonleaf no des of

T have at most one neighbor that is not a leaf no de Continue with the pro cess done

on T until the resulting partial branch decomp osition is a branch decomp osition

This algorithm is similar to Co ok and Seymour except that every nonleaf of a

partial branch decomp osition with degree greater than pro duced by the algorithm

has at most one neighbor which is not a leaf no de whereas Co ok and Seymour

allowed multiple neighbors that are nonleaf no des This algorithm is also similar to

the tree building algorithm given by Rob ertson and Seymour

E(A) v yx

E(B)

Figure The initial split

E(X) yx E(Y) a

a e

Figure Subsequent splits if jX j

How do es one nd a go o d separation Let heS denote a hyperedge where the

ends of the hyperedge are S where S is a set of no des For a nonleaf no de v in a partial

branch decomp osition let D denote the set of edges incident with v Dene H as

the hypergraph constructed from the union of hyperedges hemide where e D

So if v was the nonleaf no de of T thenH would corresp ond to G A partial branch

E(X) y E(Y) a

a e

Figure Subsequent splits if jX j

G for every decomp osition T of a graph G is called extendible if H

nonleaf no de v V T This follows from the fact that if every H had branchwidth

at most some number k then one could use the optimal branch decomp ositions of the

hypergraphs to build a branch decomp osition of G whose width is at most k One

can see that this denition of extendible is synonymous with the denition given by

Co ok and Seymour Even though T is extendible there is not a waytocheck

if a partial branch decomp osition is extendible A separation is called greedy if the

next partial branch decomp osition created by the use of the separation is extendible

if the previous partial branch decomp osition is extendible Some greedy separations

will b e describ ed along with pro ofs that they are indeed greedyLetGF denote the

subgraph of graph G induced by F a subset of no des or edges

Theorem Let G b e a graph with partial branch decomp osition

e a nonleaf no de with degree greater than T Let a V T b

i i

Let e b e the edge incidentto a and not incident to a leaf no de Let G be

the induced subgraph of G pro duced bytheleaves adjacenttoaLabel

the no des that are in the middle set of e as linking no des in G Letf

denote an edge of G that is only incident with linking no des of G Then

a a

G f G n f is a greedy separation

a a

Pro of To distinguish b etween a V T anda V T denote them

i i

as a and a resp ectively Let T b e the partial branch decomp o

i i i

sition created from T by creating a no de y and an edge a y with

i i

y having the leaves corresp onding to the edges of G n f Let T bean

optimal branch decomp osition of H Create T from T by deleting the

leaf that is mapp ed to f and for the no de of degree contracting one of

the no des edges Let b e the restriction of to T Thus T isa

y y a a

i i

branch decomp osition of H So H H and notice that H

has branchwidth at most jmidejThus G f G n f is greedy a

a D

Theorem Let GT G aand e b e dened as they were in

i a

Theorem with the addition that G is biconnected Let v V G be

a linking no de that is only incident to one edge say f and the other end

w is not a linking no de Then Gf G n f is a greedy separation

Pro of Since G is biconnected then the number of linking no des of G is

greater than one To distinguish b etween a V T anda V T

i i

denote them as a and a resp ectively Let T b e the partial

i i i

branch decomp osition created from T by creating a no de y and an

ves corresp onding to the edges of G n f edge a y with y having the lea

a i

Let T b e an optimal branch decomp osition of H Create T from

T by deleting the leaf that is mapp ed to f and for the no de of degree

contracting one of the no des edges Let b e the restriction of to T If

one replaces v with w in every middle set of T that contained v then

the resulting branch decomp osition would b e a branch decomp osition of

y y a a

i i

H Thus H H Also notice that H has branchwidth at

most jmidejThus Gf G n f is greedy

a D

Theorem Let GT G aande b e dened as they were in

i a

Theorem Supp ose G n mide has at least twocomponents Let W

b e a comp onentof G n mide Let S mide denote the linking no des

that are adjacent to at least one no de of V W inG ThenG V W

a a

S G n V W E G V W S is a greedy separation

a a

Pro of To distinguish b etween a V T and a V T denote

i i

them as a and a resp ectively Let T b e the partial branch de

i i i

comp osition created from T by creating no des x and y and edges

a x and a y with x having the leaves corresp onding to the edges

i i

of G V W S andy having the leaves corresp onding to edges of G n

a a

V W E G V W S Let T b e an optimal branch decomp osi

tion of H Create T by deleting from T the leaves that are not mapp ed

de of degree contracting one to edges in H V W S and for every no

of the no des edges Let b e the restriction of to T SoT is

a branch decomp osition of H V W S Thus the width of T is

at most the the branchwidth of H A similar argument can b e done for

y x y a a

i i

H So H H H Also notice that H has branchwidth

at most jmidejG V W S G n V W E G V W S is a

a a a separation

greedy D

Other than greedy separations one would want separations X Y such that

maxmida xmida y is minimized Co ok and Seymour used a heuristic

called the eigenvector method to nd separations for their branch decomp osition co de

The eigenvector metho d was implemented along with a metho d called the diameter

method

Eigenvector Metho d

Let G T a e and G b e dened as they were in the previous section One would like

i a

to nd a separation X Y ofG such that the following partial branch decomp ositions

are likely to have greedy separations One p ossible wayis to nd X Y such that

jX j jY j jE G j for some Also it would b e helpful if jX Y j is small

as p ossible This is a well known problem

Problem THE SEPARATOR PROBLEM Oneisgiven a graph

GV E and wishes to choose a partition X Y of E with jX j jY j

jE j and with jX Y j as small as p ossible where

The separator problem is also NPcomplete Has anything b een gained The answer

is b elieved to b e yes The separator problem can b e approximated by a heuristic

called the eigenvector metho d as follows

Let L be a jE G j by jE G j matrix Foranodev letadj v denote the set of

a a

edges incidentto v and let deg v denote the degree of v For e f E G let

if e f

Le f

deg u u V and e f adj u if e and f share an end

otherwise

the eigenvector of L corresp onding to the second smallest eigenvalue Compute

of L Toachieve this goal lo ok at the matrix N cI L where c is some large

constant and I is the identitymatrix So in order to compute the eigenvector of

L corresp onding to the second smallest eigenvalue of L one would have to compute

the eigenvector of N corresp onding to the second largest eigenvalue of N Since the

vector of all s denoted by corresp onds to the largest eigenvalue of N use the

power metho d with a vector y suchthat y

Let x b e the eigenvector of N corresp onding to the second largest eigenvalue of

N Sort the edges of G so that the numb ers x are in nondecreasing order Let A be

a i

the d jE G je rst terms and B b e the d jE G je last terms for some

a a

Given A B E G compute a partition X Y of E G such that A X

a a

B Y and jX Y j is minimized This pro cedure is done by a standard network

ow problem and is solved as follows Let H b e a minor of G with V A identied

to no de v and V B identied to no de v Next nd the smallest vertex cut in H

A B

separating all internally disjoint v v paths denoted by sepv v The idea of

A B A B

using eigenvectors to nd separators comes from the work of Alon and Chung

Diameter Metho d

The diameter ofagraphG is the smallest number k such that the shortest path

between anytwo no des of G has distance at most k where the edges all have length

equal to one The eccentricity of a no de v is the smallest number k such the shortest

path b etween v and any other no de w has distance at most k where the edges of the

graph all have length equal to one Let G b e a graph with partial branch decomp osi

tion T constructed using the construction discussed in Section Let a denote

a nonleaf no de of T with degree greater than By the tree building construction

of T a is the only nonleaf no de or a has exactly one neighbor that is also a

the edge b etween a and its nonleaf no de neighbor if a nonleaf no de Let e denote

has a nonleaf no de neighbor Let G denote the subgraph of G given by the edges

corresp onding to the leaves adjacentto aIfe exists lab el the no des of mideas

linking no des in G Again the separator problem needs to b e solved The diameter

metho d pro ceeds as follows First let the no des of G that have eccentricity equal

to the diameter of G and those that are lab eled linking no des b e lab eled as source

no des Let v b e a source no de and let b e some numb er suchthat Sort

all of the no des such that their distance to v is in nondecreasing order Let A be the

djVG je rst terms and B b e the djVG je last terms

a a

Given A B V G compute a separation X Y ofG such that A V X

a a

B V Y and jV X V Y j is minimized This pro cedure is done by a standard

network ow problem and is solved as follows Let H beaminorofG with A

identied to no de v and B identied to no de v Then nd the smallest vertex cut

A B

in H intersecting all v v paths denoted by sepv v Let us lab el the no des of

A B A B

sepv v inG as separation no des and denote the cardinalityby new Let the

A B a

no des of G that are b oth lab eled linking no des and separation no des also b e lab eled

share no des and denote their cardinalityby sh Lab el the linking no des that are on

one side of the cut but are not lab eled separation no des as side no des and denote

their cardinalityby s Let l ink ing denote the cardinality of linking no des in G Let

work b e dened as maxs new l ink ing s sh new and dene play

as mins new l ink ing s sh new So there is a particular work and

play for the pair v The metho d iterates over all source no des and dierentvalues

of is initially used and the cardinality of To iterate over the dierentvalues

of A and B is decreased by some parameter t to nd other values of This metho d

nds the pair v such that the separation pro duced by them has the smallest

o o

work denoted by ow or k and the biggest play out of all the pairs v with the same

work denoted by opl ay So the pair v pro duces a separation of G such that

o o a

maxmida xmida y is minimizedand minmida xmida y is maximized

and the sets of ow or k and opl ay will b e the middle sets of the two new edges created

in the next partial branch decomp osition

To illustrate the diameter metho d let Figure represent G with the linking

no des of G colored black So the source no des of Figure are the linking no des and

of no des i and k then no des i and bIfA consisted of no des a and b and B consisted

0 1

b 45

8 d e f

10 11 i

14 15

Figure Example G

sepv v would consist of d eandf The side no de would b e a and the share no de

A B

would b e f Thework of source no de a with equal to is four and the playis

also four Figure illustrates what G and G would lo ok like if the pair a is

x y

used as the b est pair to give us a separation The basic premise is that if the shortest

path in the graph G between no des u and v is the diameter and the diameter is large

then edges incident with u should not b e close in a branch decomp osition to edges

incident with v

Test Instances

The test instances are partitioned into three classes planar compiler and llin The

graph tele under the planar class is a no de graph from the telecommunications

industry The rest of the planar test instances are Delaunay triangulations of some

of the graphs from the TSPLIB library of standard test instances for the traveling

salesmen problem For more information ab out Delaunay triangulations the reader is

referred to Edelsbrunner Figure illustrates some of the planar test instances

with their branchwidth

0 1 Gx b 45

8 f d e

f d e 10 11 i G 14 15 y

Figure Example G and G

x y

tele133 pcb442

5 17

Figure Planar Test Instances with their branchwidth

The compiler test instances are instances of controlow graphs from actual C com

pilations These test instances were provided by Keith Co op er Asso ciate Professor

of Computer Science at Rice University As stated earlier the controlow graph of

some program P is a digraph but the orientation of the edges is unimp ortant for

coloring So the graphs are p erceived as undirected Most of the instances are series

parallel graphs but there are a few that have branchwidth at least and are not

planar Figure illustrates some of the graphs in this class and b ounds on their branchwidth

trans rkfs

4 at most 5

Figure Compiler Test Instances with their branchwidth

The llin test instances are instances generated by matrices These instances

were provided byChao Yang researcher for Oak Ridge National Lab oratory Let

M be an n by n matrix for some n M can b e mo deled as a graph in which

the no des represent the dierent columns and rows of M andanedgebetween no de

i and no de j represents if M i j is nonzero Manyof theinteresting matrices for

minimum llin are huge more than no des however the smaller cases were

only considered Figure illustrates some the graphs in this class and b ounds on

their branchwidth

bcs03 bcs01

3 at most 12

Figure Fillin Test Instances with their branchwidth

Computational Results

All computations were p erformed on a MHz Sun UltraSparc using co de written in

the C programming language The branchwidth of each graph was calculated using

an implementation of the Seymour and Thomas branchwidth algorithm for planar

graphs NP under the branchwidth column means that the graph was nonplanar

and the branchwidth of that graph is unknown The diameter metho d was compared

to the eigenvector metho d due to Co ok and Seymour and the resulting widths of

the eigenvector metho d are under the eigenvector column while the resulting widths

of the diameter metho d are under the diameter column Under the eigenvector

column the eigenvector metho d had the restriction of only working on biconnected

graphs So an NB in this column means the graph was not biconnected and a

branch decomp osition was not computed for this graph using the eigenvector metho d

The hybrid column represents a hybrid of b oth heuristics where the eigenvector

metho d is used on the initial separation and the diameter metho d is used on the

subsequent separation

For the class lab eled planar graphs given in Tables and the hybrid

metho d has promising results The hybrid metho d did not compute an optimal

branch decomp osition for only of the planar graphs Furthermore of the

branch decomp ositions have widths only one more than the branchwidth of those

graphs Similarly the diameter metho d computed an optimal branch decomp osition

for of the graphs with only branch decomp ositions width one more than the

branchwidths for the graphs

For the classes lab eled llin and compiler given in Tables

and the diameter metho d did not give the optimal branch decomp osition for

only one graph that was planar The hybrid metho d did not give the optimal branch

decomp osition for two planar graphs The hybrid metho d gives more lower widths

than the diameter metho d for the llin graphs but the diameter metho d gave more

lower widths than the hybrid metho d for the compiler graphs

Unfortunately the eigenvector metho d used for this comparison was embedded in

another software So there is not a record of the running times of the eigenvector

metho d From observance the eigenvector metho d was somewhat faster than the di

ameter or hybrid metho d One reason is that b oth the diameter and hybrid metho d

search through a numb er of separations b efore concluding to use one of those separa

tions for tree building however the eigenvector is designed to pro duce one separation

to b e used for tree building

Table Planar graphs

diameter hybrid

planar graphs width time sec width time sec eigenvector branchwidth

tele

atspdel

biertspdel

chtspdel

dtspdel NB

dtspdel

dtspdel NB

eiltspdel

tspdel

giltspdel

kroAtspdel

kroBtspdel

kroCtspdel

kroEtspdel

lintspdel

nrwtspdel NB

ptspdel

p cbtspdel NB

p cbtspdel

Table Some more planar graphs

diameter hybrid

planar graphs width time sec width time sec eigenvector branchwidth

prtspdel NB

prtspdel

prtspdel NB

prtspdel

rattspdel

rattspdel NB

rattspdel

rdtspdel

rltspdel NB

tsptspdel

utspdel NB

utspdel

utspdel NB

utspdel

vmtspdel

vmtspdel NB

Table Compiler graphs

diameter hybrid

comp graphs width time sec width time sec eigenvector branchwidth

abrt NB

aclear NB

advbnd NB NP

arret

bcnd

bcndb

bcndl

bcndr

b cndt

bilan

bilsla NB

blts NB

buts NB

cardeb

celbnd NB NP

clrdt NB

co eray

colbur NP

cosqb NB

cosqb

cosqf NB

cosqf

cosqi NB

dco era

ddeu

debu NB

debico NB

decomp NB NP

denitl NB

denitr NB

denpt NB

dens

densx

densy

deseco NB

diagns NB NP

drepvi NB

drigl

dyeh

ecrd NB

ecwr NB

ell NB

endrun NB

energy NB

erf NB

Table More compiler graphs

diameter hybrid

comp graphs width time sec width time sec eigenvector branchwidth

erhs NB

error NB

exact NB

fehl NB

tb NB

tf NB

eld NB NP

fmin NB NP

fmtgen NB NP

fmtset NB

fpppp NB

gamgen NB

genb

genprb

getb NB

getdt NB

heat NP

hmoy

ibin

ihbtr NB

ilsw NB

inibnd NB

inideb NB

iniset NB

inisla NB

init NB NP

inithx NB

injall NB

injbat NB

injchk

injcon NB NP

integr NB NP

inter

intowp NB

jacld NB

jacu NB

jobtim NB

lnorm NB

lasden NB NP

laser

lasp ow

lclear NB

linj NP

lissag NB

main NB

Table More compiler graphs

diameter hybrid

comp graphs width time sec width time sec eigenvector branchwidth

maxnorm NB

nprio

numb NB

orgpar NB

parmov NB

parmve NB

parmvr NB

paroi NB

pastem

pdiag NB

pintgr NB

prophy

putb NB NP

putdt NB

radb NB NP

radbg NB NP

radf NB NP

radfg NB NP

ranf NB

recre NP

repvid NB

rewdt NB

rtb NB

rtb

rtf NB

rtf

rti NB

rti

rhs NB

rinj NP

rkf NB

rkfs NB NP

saturr

saxpy NB

setb NB

setbv NB

setinj

setiv NB

seval NP

Table More compiler graphs

diameter hybrid

comp graphs width time sec width time sec eigenvector branchwidth

sgemm NB

sgemv NB

sigma NB

sinqb NB

sinqf NB

sinqi NB

slvxy NB

smo oth NB NP

solvy NB

solve NB

sortie NP

spline NB

ssor NB

subb NB

sudtbl

supp

svd NB NP

tcomp NP

tpart

trans NB

twldrv NB NP

urand NB

vavg

verify

vg jyeh

vnewl NB

vslvp NB NP

vslvx NB NP

xy NB

yeh

zeroin NB NP

Table Minimum Fillin graphs

diameter hybrid

llin graphs width time sec width time sec eigenvector branchwidth

b cs NP

b cs NB

b cs NB NP

Chapter

Planar Branch Decomp ositions

Since many of the test instances are planar the algorithm of Seymour and Thomas

to nd the branchwidth of planar graphs was implemented for comparison with the

general branch decomp osition heuristics In this work the algorithm prop osed by

Seymour and Thomas for the the branchwidth of planar graphs will b e referred to as

the ratcatcher method and will b e discussed in Section Seymour and Thomas

also oered a p olynomial time algorithm using the ratcatcher metho d to compute op

timal branch decomp ositions for planar graphs which will b e referred to as the edge

contraction metho d In Section a practical implementation of the edge contrac

tion metho d by adding a few heuristics is oered This metho d the cycle method

improves on the edge contraction metho d by using a divide and conquer approach

Ratcatcher Metho d

Consider a twoplayer game played on a planar graph Think of the graph as a o or

plan of a house where the regions are ro oms and the edges are walls also assume

that eachwall has a corresp onding do or One player of the game is a rat that is

content on staying in the house and running along the walls of the house The other

player is a ratcatcher who was hired bytheowners of the house to catch the rat The

ratcatchers only weap on to catch the rat is a whistle of some xed sound level k

Dep ending on the thickness of the walls the whistles sound can p enetrate the walls

aecting the rat the rat can not run along noisy walls The ratcatcher can catch

the rat if the rat is cornered by noisy walls The rules of the game are that each

player knows the lo cation of the other and once the ratcatcher enters a do orwayon

one move the ratcatcher has to go through the do orwayinto the next ro om on the

next move Seymour and Thomas showed that the largest sound level such that

the rat will always escap e is asso ciated with the structure of the planar graph that

will b e dened in the following section Below the foundational terms and concepts

needed to explain the ratcatcher metho d are given

Carvingwidth

Let H b e a graph with no de set V H and edge set E H Let T b e a tree having

jV H j leaves in whichevery nonleaf no de has degree Let b e a one to one

function from the no des of H to the leaves of T The pair T is called a carving

decomposition of H Notice that removing an edge of T say e partitions the no des of

H into two subsets A and B Thecut set of e is the set of edges that are incidentto

e e

no des in A and to no des in B The width of a carving decomp osition T isthe

e e

maximum cardinality of the cut sets for all edges in T Thecarvingwidth for H is the

minimum width over all carving decomp ositions of H The ratcatcher algorithm is

really an algorithm to compute the carvingwidth for planar graphs In order to show

the relation b etween carvingwidth and branchwidth another denition is needed

Let G b e a planar graph Take a drawing of G on a sphere Let M b e a graph

with the vertex set E G and let C v V G b e a collection of circuits of M with

the following prop erties

the circuits C v V G are pairwise edgedisjoint and have union E M and

for each v V G let the edges of v in G be enumerated according to the cyclic

x then C has vertex set fx x g and in C x is order of v ie x

t v t v ii

adjacentto x i t where x means x

i t

Then M is called a medial graph of G The medial graph of Figure is given in

Figure

0 a b

2 4

ef3

1 5 6 7

gh8

10 9

cd11

Figure Q from Chapter

4 C C a 2 b

CCef3

1 6 75

8 C C g h

C c 910C d

Figure Medial graph of Figure

Seymour and Thomas proved

Theorem Let G b e a connected planar graph with jE Gj and

let M b e the medial graph of G Then the brachwidth of G is half the

carvingwidth of M

So computing the carvingwidth of M also gives us the branchwidth of G The

algorithm really do es not searchfor low cut sets but searches for ob jects that prohibit

the existence of low cut sets The authors call these ob jects antipodalities

Antipo dality

Given a planar graph G and its dual G an edgee of G is incident with a region

r V G ife is incidentto r Awalk in a G is a sequence v e v e e v where

t t

v v v V G e e e E G and v v is the set of ends of e i t

t t i i i

Awalk is closed if v equals v Anantipodality in G of range at least k is a function

with domain E G V G suchthatforalle E G e is a nonnull subgraph

of G and for all r V G r is a nonempty subset of V G satisfying

A If e E Gthen noendof e b elongs to V e

A If e E G r V G and e is incidentto r then r V e and

every comp onentof ehas a vertex in r

A If e E Gandf E e then every closed walk of G using e and

f has length k

The main result in Seymour and Thomas is the following

Theorem Let G b e a connected planar graph with jV Gj and

let k bean integer Then G has carvingwidth at least k if and only

if either v k for some no de v or G has an antip o dality of range at

least k

So an antipo dality of range at least k is an escap e strategy for the rat when the

sound level of the ratcatchers whistle is k When the ratcatcher is in the do orwayof

some edge e the rat knows the ratcatcher will visit ro om r So the rat travels along

the edges of antipo dalityof e until it reaches a no de in the antipo dalityof r Ifthe

rat follows this pro cedure the rat will never b e caught

The Algorithm

Given a graph G the ratcatcher algorithm constructs M and M and tests if the

carvingwidth of M is at least k by testing if there exists an antipo dality of range at

least k To test for the existence of an antipo dality of range at least k for an edge

e of the medial graph edges are indentied that satisfy A and A For each

e E M let e b e the subgraph of M consisting of all the edges f such that no

end of f is an end of e and no closed walk of M of length less than k contains b oth

e and f The authors proved that if there exists an antipo dalityof M of range at

least k then there is one say such that e is a union of comp onents of e for

each e E M D the matrix that stores allpairs shortest paths of M was

computed to satisfy the closed walk condition for e

To test for the existence of an antipo dality of range at least k for a region r of

the medial graph take the largest set of no des p ossible then test if at least one of

the no des in the set satises condition A To get the largest set p ossible for the

antipo dalityof r letr betheintersection of V M n R where R is the set of

no des of M that are incidentto edges e incident to region r In order to satify A

completelya node v in r s antip o dalityhas tohave a path on the edges of e from

v to some no de w where w is in es other regions antipo dality for all edges e incident

with r If a no de do esnt satisfy A for r then delete the no de from r If initially

r isthenull set or if r b ecomes the null set then an antipo dality of range at

least k do es not exist

Antipo dalityk M M D

Input M M D k

Output yes or no

For each edge e of M

a Lab el the edges of e

For eachnode r of M

a r V M n R where R is the set of no des of V M incident to region r

b If r isnull set then return no

While r V M whose cardinalityof r has changed or has not b een tested

using comptest

a For r V M

i If comptestr returns no then return no

Input e M M D k

Output yes or no

For each edge f of E M

a If f has no ends equal to an end of e

vw and f xy i Let e

ii If D v xD w y k and D v y D w x k

A Lab el f as part of e

Comptestr

Input r M

Output yes or no

For each edge f incidentto r

Let or denotes f s other region

a For eachnodev r

i If there do es not exists a path on the edges of f from v to a no de

in or then delete v from r

If r null set then return no

Else return yes

Input planar graph G

Output branchwidth

Construct M and M

Compute all pairs shortest paths on M and store in matrix D

Let max the maximum degree of M

If max is o dd

a Let k max

b Let ol dk max Else

a Let k max

b Let ol dk max

antip o dalityk M M D

While an antipo dalityofrange k

a ol dk k

b k k

c antipo dalityk M M D

Return ol dk

Cycle Metho d

Recall from Section that a partition of the edges of the original graph corresp onds

to an edge cut in the corresp onding medial graph Also recall that the medial graph

of a planar graph is also planar In fact Rob ertson and Seymour state that a

medial graph of a planar hypergraph is also planar The cycle metho d uses this fact

and the fact that b onds in an embedded planar graph corresp ond to cycles in the

corresp onding dual of the embedding Throughout this section graph could b e

replaced byhyp ergraph but hypergraph can not b e replaced by graph Also

assume that the same tree building techniques discussed in Chapter are used For

the initial split let C beacycleinM G of length twice G C corresp onds to

A B a separation of G In order to check if this separation can b e used for an

adding a hyp eredge with optimal branch decomp osition create hypergraph H by

ends V A V B to A Rep eat the pro cedure to pro duce H If the branchwidth

of H and H are at most the branchwidth of G then A B can b e used for an

A B

optimal branch decomp osition An augmenting path algorithm by Suurballe and

Suurballe and Tarjan is used to compute the desired cycles in M G Since there

exist separations of G that have order equal to the branchwidth of G but are not found

in any optimal branch decomp osition of G and there is no algorithm guaranteed to

nd all cycles of a graph with a particular length one relies on the edge contraction

metho d if no cycles were found that could have b een used in an optimal branch decomp osition

a he 2 3 c 6 7 9 f g h 12 13 j 16 17 k

Figure Example hypergraph H

By the tree building construction of Section there exists a nonleaf no de v

that has only one neighbor w that is not a leaf Let e b e the edge b etween v and w

Let G b e the subgraph of G asso ciated with the leaf neighb ors of v LetH b e the

v v

hypergraph created by adding a hyperedge with ends midetoG Figure is an

example of H if v was adjacent to leaves corresp onding to edges

and for Figure and the hyperedge of Figure is lab eled he Figure is

the medial graph of Figure and Figure is the corresp onding dual of Figure

with the outside region representing the hyperedge Also the regions of Figure

other than the outside region are lab eled corresp onding to the no des of Figure

Let C E M H denote the cycle that b orders the face corresp onding to the v

r1 2 r2 r4 r3 67 he r6 r7 r8 r5 9

r9 12 13 r10 r11 16 r12

Figure Medial graph of Figure

hyperedge C is illustrated in Figure by the dashed cycle The cycle metho d nds

the shortest path b etween s t V C using only the edges of E M H n E C

which are the solid edges of Figure such that the three cycles created bythe

path and C eachhavelengthat mosttwice the branchwidth of G For example

let P denote the path r r r r r illustrated by thick edges in Figure

Notice that P C creates three cycles and each cycle has length at most Also

notice that P actually partitions the regions and from the other lab eled regions

of Figure and this corresp onds to a separation G G nf g Figure

v v

is an illustration of H and H corresp onding to the aforementioned separation In

A B

summary the cycle metho d creates M H and nds a path in E M H n E C to

v v

the length constraint on the resulting cycles and the branchwidth constraint satisfy

on the corresp onding separation

Let us estimate the running time of the metho d First let us consider the running

time to compute a separation for subsequent splits Let G b e the original graph

and dene m jV M Gj jE M Gj As one can see it will takeat most

317 9

r2 r7 r12 7 13 r4 r6 r10 2 6 12 16

r1 r3 r5 r9 r11

Figure Corresp onding dual graph of Figure

O M Gm time to nd all p ossible paths b etween anytwo no des on C Since

Seymour and Thomas state that the ratcatcher metho d takes time approximately

O m to compute the branchwidth of a planar graph the algorithm takes time

approximately O M Gm to compute a separation Without loss of generality

one can assume that G is lo opless b ecause lo ops do not aect the branchwidth of

the graph Since G is lo opless one can easily see that m e and let e jE Gj

Since the metho d is a divideandconquer algorithm then the approximate run time

is O Ge log e for subsequent splits For the initial split the augmenting path

algorithm has complexity O e just to compute one cycle So the initial split takes

complexity O e and the total complexity to compute optimal branch decomp ositions

using the cycle metho d is O e

Edge Contraction Metho d

Now that the cycle metho d has b een dened a comparison of the cycle metho d to the

edge contraction metho d oered by Seymour and Thomas is in order In order

a H A 2 3 c h c 6 7 9 h H f g B 12 13 j 16 17 k

Figure Resulting hypergraphs from using path P

to pro ceed with the comparison the edge contraction metho d must b e dened The

structure of the edge contraction metho d allows nonleaf no des in a partial branch

decomp osition to have more than one nonleaf neighb or which is similar to the tree

building techniques of Co ok and Seymour This corresp onds to H s with multiple

hyperedges The metho d pro ceeds as follows

Compute M H

For each uv E M H

a If M H nfu v g is connected

i Let M b e obtained from M H bycontracting all edges b etween u

o v

and v

ii If M has carvingwidth G then Gu v G nfu v gis the

new separation

Seymour and Thomas state that the complexity for the edge contraction metho d

to compute an optimal branch decomp osition takes approximately O e equivalent

to the worst case time for the cycle metho d

Computational Results

The test instances are the planar class of test instances of Section Figures

and illustrate some of the test instances Both metho ds were implemented using

the C language All computations were p erformed on a MHz Pentium Pro

machine with megabytes of memoryTable compares the edge contraction

metho d to an unaltered version of the cycle metho d lab eled cycle and to the

cycle metho d with the addition of using the greedy separations dened in Section

From the results given in Table the cycle metho d is an improvement to the edge

contraction metho d One explanation of this observation is that the number of edges

of the graphs at each iteration of the edge contraction metho d decreases by one while

the cycle metho d is more of a divide and conquer type algorithm Even though

the worst case analysis of the metho ds are the same the cycle metho d nds optimal

branch decomp ositions faster than the edge contraction metho d by factors ranging

from to Table is a table of the running times of the cycle metho d with

the addition of using greedy separations on some larger test instances These test

instances were to o large for the edge contraction metho d to handle in a reasonable

time frame seconds In addition the cycle metho d never needed to rely on

the edge contraction metho d for a separation in any of the test instances

tele133 ch130

5 10

Figure Some Test Instances with their corresp onding branchwidth

rd400 pcb442

17 17

Figure Some Test Instances with their corresp onding branchwidth

Planar Graphs and Their Duals

After understanding the concepts and terms in Seymour and Thomas a rela

tionship b etween the branchwidth of a particular class of planar graphs and their

corresp onding duals was discovered Recall for a planar embedding H of a graph G

M H the medial graph of H isdenedby V M H E G and for each a V H

there exists a cycle around the edges incidentto a in E M H corresp onding to the

cyclic order of a in H Below the main result of this section and a pro of are given

Theorem Let G b e a connected planar graph and let G b e the

corresp onding dual suchthatG and G are lo opless then G G

u574 d657

17 22

Figure Some Test Instances with their corresp onding branchwidth

Pro of Let H b e a planar embedding of G and let H b e the corre

sp onding dual of H

Claim ef E M H if and only if e f E M H

Pro of Since ef E M H and H is lo opless then a

V H suchthate and f are consecutivein as cyclic order By

the denition of a dual graph and the fact that H is lo opless

b V H such that e and f are consecutivein bs cyclic

order e f E M H By denition of a dual graph the

is satised

claim D

een V M H By the denition of dual graphs there is a bijection betw

and V M H By the claim uv E M H if and only if uv

E M H Thus M H M H Since M H andM H are

isomorphic the carvingwidth of M H is equal to the carvingwidth of

H By Theorem G G

M D

Given a planar graph G itisknown that edge deletions in G corresp ond to edge

contractions in G and vice versa So the following corollary can b e useful in

discovering the obstruction sets of the minor closed classes of graphs characterized

by b ounded branchwidth

Corollary Supp ose that G is a lo opless planar graph with a lo opless

dual G and that G is in the obstruction set of the class of graphs with

branchwidth at most k where k Thus G is also in the obstruction

set

One can notice that M and Q in Figure are duals of each other and they b oth

have branchwidth equal to Also Theorem implies a stronger result whichis

Theorem Let G b e a planar embedding of a planar graph with

corresp onding dual G such that G and G are b oth lo opless Let T

b e a branch decomp osition of G Then there exist a branch decomp osition

T such that if v is a leaf of T then v v and the width of

T is equal to the width of T

Figure is drawing of M with the edges lab eled corresp onding to the edges of Q

drawn in Figure Figure and Figure are an example of this theorem

a 2 3 78

05 6 10

{b, d, e} {c, d, f} 1 4 11 b c {b, c, e, f}

37 {a, b, c} {a, e, f} 10 2 d 5 11 6 8

e f 0419

Figure M has branchwidth

Table Cycle Metho d versus Edge Contraction Metho d

Graphs Edges G G sec EC sec Cycle sec Cycle greedy sec

eiltspdel

tele

eiltspdel

prtspdel

rattspdel

kroEtspdel

prtspdel

kroBtspdel

kroAtspdel

rdtspdel

kroCtspdel

eiltspdel

lintspdel

prtspdel

biertspdel

htspdel c

prtspdel

utspdel

kroAtspdel

chtspdel

kroBtspdel

rattspdel

dtspdel

kroBtspdel

prtspdel

kroAtspdel

tsptspdel

prtspdel

giltspdel

atspdel

prtspdel

tspdel

rdtspdel

p cbtspdel

prtspdel

rattspdel

utspdel

ptspdel

dtspdel

Table Cycle Metho d Results

Graphs Edges G G sec Cycle greedy sec

utspdel

rattspdel

vmtspdel

prtspdel

utspdel

cbtspdel p

dtspdel

rltspdel

nrwtspdel

tspdel

utspdel

tspdel

vmtspdel

dtspdel

utspdel

rltspdel

dtspdel

utspdel

prtspdel

Chapter

Characterizing Branchwidth

Rob ertson and Seymour characterized the branchwidth of graphs with branch

width at most and the ngrid graphs which are shown in Table This chapter

is dedicated to characterizing the branchwidth of two sp ecic graph classes Halin

graphs and chordal graphs

Table Classes of Graphs and their branchwidth

Classes Branchwidth

Forests

Seriesparallel

Outerplanar

nGrid n

Halin

Chordal d Ge G G

Halin Branchwidth

A Halin graph can b e constructed from a tree whose nonleaf no des have degree at

least three embedded in the plane by adding a circuit through the leaves of the tree

according to their cyclic order imp osed by the embedding From this description of

Halin graphs one can derive that they are connected and planar One can also

derivethat K is the only minor minimal Halin graph From this observation and

Theorems and one can deduce that all Halin graphs have branchwidth at

least The ob jective of this section is to prove that all Halin graphs even the one

illustrated in Figure have branchwidth equal to

Theorem Let G b e a Halin graph then G

Pro of Let T b e the tree that was used to create G No des referenced in

T will synonymous to their corresp onding no des in GLetb b e a nonleaf

no de of T LetT b e an initial partial branch decomp osition describ ed

in Chapter Ro ot T at b and let a and c b e the leftmost and rightmost

children of a subtree ro oted at some no de d The no des a bandc will

give an initial separation of G for T to b ecome T Find greedy

separations until there are none remaining and let T b e the resulting

partial branch decomp osition If T isabranch decomp osition then

stop by the initial separation and greedy separations the width of T

is If T is not a branch decomp osition then by the design of T

k k

T has a nonleaf no de v who has only one nonleaf neighbor Let e b e the

edge in T be tween v and v s nonleaf neighbor and let the no des of this

middle set b e lab eled rt l andr where rt is a nonleaf no de in T Let

v v

G b e the subgraph of G induced by the leaf neighbors of v and let T be

corresp onding subtree of T Since T is a partial branch decomp osition

and there are no remaining greedy separations then rt is the ro ot of T

with l and r b eing in two dierent subtrees where l is the leftmost child in

its subtree and r is the rightmost child in its subtree Let lr b e the right

most child in the subtree with l and denote the subgraph of G induced by

the no des of that subtree as LThenL G n L is a separation and use

this separation to construct T Let us call this type of separation a

subtree separation By the construction of T T will have

k k

b ecause the middle sets of the two edges excluding e incident width

with v will b e frt l lrg and frt lrrg resp ectivelyContinue the pro cess

of nding greedy and subtree separations until a branch decomp osition

is reached The resulting branch decomp osition will have width equal to

Figure The branchwidth of Halin graphs is

After examining a theorem in Rob ertson and Seymour it was noted that a

generalization of the constructive pro of of Theorem could b e applied to that par

ticular theorem to give a b etter result Below necessary denitions and the improved

result are given

Let G b e a planar graph For every region of a drawing M of G dene dRto

b e the minimum value of k such that there is a sequence R R R of the regions

of M where R is the innite region and R Randfor j k there is a

vertex v incident with b oth R and R The radius M is the minimum value of

j j

d such that dR d for all regions R of M Theradius of a planar graph is the

minimum of the radii of its drawings From a generalized form of the construction

erformed in the pro of of Theorem to a pro of of Rob ertson and Seymour p

and some theorems from Chapter one can conclude that if a planar graph G has

a radius at most d for some p ositiveinteger d then the branchwidth of G is at most

d and the treewidth is at most d

Chordal Branchwidth

Some NPcomplete problems b ecome p olynomial time solvable if the problem is mo d

eled on chordal graphs For example the chromatic number problem and the min

imum llin problem are p olynomial time solvable on chordal graphs A graph is

chordal if and only if every cycle with length four or more has an edge b etween two

nonconsecutive no des in the cycle For example the complete graphs are chordal

Figure H

For an arbitrary graph G let G denote the largest clique that is a subgraph of

G Below is a theorem that characterizes the branchwidth of chordal graphs

Theorem Let G beachordal graph with at least no des then

d Ge G G

Pro of By the denition of G K is a minor of GSoby Theorem

the lower b ound is satised Bo dlaender states that the treewidth of

achordal graph is G So by Theorem the upp er b ound is

D satised

Any complete graph with at least no des satises the lower b ound One task for

this thesis was to nd a class of graphs that satises the upp er b ound Let k be any

integer greater than Start with K the complete bipartite graph on k no des

Delete a onefactor from K to pro duce M SinceM is bipartite then let A B

kk k k

b e the bipartition of V M Add edges to M such that the induced subgraph of M

k k k

from A is complete Denote the resulting graph as H Figure and Figure are

examples of H and H This section will conclude with a pro of that for k H

satises the upp er b ound of the inequalitygiven in Theorem

Figure H

Theorem H is chordal and the branchwidth of H is equal to

k k

H whichisk

Pro of First H is shown to b e chordal Let C b e a cycle of length

at least in H Since A induces a clique one need only consider the

case when there exists a member w of B on C By the construction of

H w has two neighbors u v on C suchthat u v A Since C has

length at least and A induces a clique then C hasachord Thus H is

chordal By the construction of H H isk Also by the construction

k k

of H the graph K minus a one factor is a minor of H Rob ertson

k kk k

and Seymour state that the branchwidth of K minus a one factor

where k is greater than is k Therefore k H by Theorem

Also H k by Theorem Thus H k k k D

Chapter

Branch Decomp osition Applications

Since the ma jor motivation for the construction of branch decomp ositions is their use

in solving instances of NPcomplete problems in p olynomial time this work would not

b e complete without a chapter describing some applications of branch decomp ositions

Two problems in particular that will b e discussed in this chapter are the minimum

llin problem and the minor containment problem

Minimum Fillin

Bo dlaender and Bo dlaender Kloks Kratsch and Muller use tree decom

p ositions to solve the minimum llin problem The minimum llin problem is

the problem of nding an order to p ermute the rows and columns of a matrix for

Cholesky factorization such that the creation of new nonzero entries in the matrix is

minimized An elimination ordering for a graph G is an ordering of the no des

of G such that as no de v is deleted from some corresp onding graph G then edges

i i

are added to G suchthat v s neighb ors induce a clique in the resulting graph G

i i i

The problem can b e mo deled on a bipartite graph where the rows and columns of

the matrix corresp ond to no des in the graph and the edges of the graph corresp ond

to nonzero entries of the matrix So the problem is equivalent to nding an elimina

tion ordering of the no des of the graph such that the total sum of new edges in the

resulting graphs is minimized

Since the problem can b e mo deled on a graph a heuristic using branch decom

p ositions to solve the minimum llin problem is presented First ro ot the branch

decomp osition of the graph and transform the tree into a binary tree by sub dividing

an edge and ro oting the newly created no de Visit the no des of the tree in p ost depth

rst search order Once a no de v is not contained in the middle set of the parent

edge for some parent no de in the tree but is contained in one of the child edges of the

parent no de then input v into the elimination ordering

Table reveals the results of this heuristic bw compared with the most common

heuristics used for this problem which are multiple minimum degree MMD no de

separator nested dissection no de ND and edge separator nested dissection edge

ND used under the free software package METIS oered by Karypis The

numb ers in Table under the heuristics are the actual number of extra edges that

would have b een added during the elimination ordering From Table one can see

that the branchwidth heuristic is comp etitive with the other heuristics and only did

not generate the lowest number on two instances where the actual branchwidth of the

graphs were not known and could b e considerably lower than recorded

Table Minimum Fillin Results

graphs no des edges bw MMD no de ND edge ND

bcso

bcs

Minor Con tainment

From Rob ertson and Seymour it is known that testing for minor containmentin

a graph with a xed minor is p olynomial time solvable but general minor containment

is known to b e NPcomplete Rob ertson and Seymour also gave a general

framework to use a branch decomp osition of a graph G to test if a graph H is a minor

of G This section details the use of a branch decomp osition to test for general minors

with the addition of some metho ds to decrease the numb er of graphs created during

the pro cess

Initially the pro cess includes making the tree of the branch decomp osition into a

ro oted binary tree and visiting the no des of the tree in p ost depthrst searchorder

as in Section But instead of nding an elimination ordering of the no des of G

each no de will corresp ond to a set of minors of Ga minor set and the algorithm

checks to see if one of the graphs in these minor sets is isomorphic to the graph H

This isomorphism testing is done by rst nding the canonical lab eling of the no des

of each graph This lab eling corresp onds to p ermuting the columns of the adjacency

matrix of the graph such that the binary number from the entries ab ove the main

diagonal written column by column has the lowest value as p ossible The algorithm

used to achieve this goal can b e found in Kreher and Stinson

In order to discuss the algorithm for minor containment further some foundational

denitions and assumptions are needed For a no de n of the branch decomp osition

every no de of every graph in the minor set of n will corresp ond to two sets of no des

of G One set will contain all the no des of G that were identied to create that no de

while the other set will only contain the intersection of the rst set with the middle

set of the parentedgeof n The rst set will b e referred as the history set of the no de

and the second set will b e referred as the linking set of the no de Also a no de with a

nonempty linking set is called a linking no de Tokeep away from redundancy in the

minor set a denition of isomorphism in terms of minor set graphs is needed Given

a isomorphism between two minor set graphs G and G G and G are called

minor isomorphic if u V G and v V G corresp ond under the isomorphism

then the linking set of u and the linking set of v must b e equivalent Assume that H

and G are simple and connected and an edge in a graph of a minor set will never b e deleted

0------0a b !

0a 0 b 0-----0 a b 0 a,b

Figure The minor set for a leaf

For the leaves of the ro oted branch decomp osition if the leaf corresp onds to edge

ab and b oth a and b are in the middle set of the leaf edge then the minor set for

that leaf includes three graphs One graph is only the edge ab while another is the

two no des with the edge deleted The nal graph is an isolated no de with linking set

fa bg An example of this op eration with corresp onding linking sets can b e seen in

Figure If the middle set only included a then the minor set for the leaf would

include a graph of an isolated no de with the linking set b eing only a and a graph

of the edge ab The graph of two isolated no des would not b e considered b ecause

it would violate the connectivity assumption The graphs created by deleting no des

are not considered for the minor sets b ecause by the connectivity assumption one

can assume that the no des that are supp osed to b e deleted can b e identied to other

no des

For a nonleaf no de p in the ro oted branch decomp osition p will havechildren l

and r Supp ose that graph G is a graph in the minor set of l and G is a graph in the

l r

minor set of r Let S represent the intersection of the middle sets of the child edges

of p and lab el the no des of G and G as intersect nodes if the intersection of their

l r

linking sets with S is nonemptyFor eachintersect no de v of G identify all of the l

l r

Figure Joining minor set graphs for a nonleaf no de p

intersect no des of G that have linking sets that contain elements in the intersection

of S and the linking set of v Denote the resulting graph as M For example if v

had fa bg as the intersection of S with v s linking set and there exists u w V G

such that the intersection of us linking set and w s linking set with S are fag and

fbg resp ectively then u and w would b e identied into one no de in M Rep eat the

pro cess to pro duce M If there is a matching by linking sets of the intersect no des

of M with the intersect no des of M then identify the no des by the matching to

r l

pro duce M a graph for the minor set of p Also up date the linking sets of the

no des of M to corresp ond to the middle set of ps parent edge Figure gives an

cess for nonleaf no des with the white intersect no des lined up illustration of the pro

with their matching no de

In order to make the algorithm ecient pruning techniques havetobediscussed

Some foundational pruning techniques are the use of H s minimum and maximum

degree and H s connectivity to prune the minor sets of unusable graphs For example

if the algorithm built a graph in a minor set whose maximum degree is greater than

H s maximum degree then that graph can b e pruned If a graph in a minor set has a

nonlinking no de with a degree smaller than the the minimum degree of H then that

graph can also b e pruned Also if a comp onent of a minor set has k no des where k is

anumb er greater than the connectivityof H but less than the numb er of no des of H

and the numb er of linking no des of this comp onent is less than the connectivity then

that graph can b e pruned Some other pruning techniques that will b e discussed in

detail in the following subsections are edges discrimination and maximum clique

Edge Discrimination

In the previous section it was discussed that sometimes intersect no des of a minor set

graph have to b e identied in order to merge with another minor set graph of another

child no de to create a new minor set graph for the parent no de Edge discrimination is

based on this identifying pro cedure Supp ose the algorithm is building a minor set for

nonleaf no de p and l r G and G are dened as they were in the previous section

l r

So G is actually a minor of a subgraph of G corresp onding to the graph induced

by the leaves of the tree under the subtree ro oted by l let us call this subgraph G

Under edge discrimination if G has two distinct intersect no des x and y that must

b e identied to create M and if there is an edge in G between twon odes s and t

suchthat s is in the history set of x and t is in the history set of y then the algorithm

would not consider joining G with G The following theorem will prove that edge

l r

discrimination will not prune away a minor set graph that is isomorphic to H

Theorem Edge discrimination will not prune away a minor set

graph that is isomorphic to H

Pro of From previous assumptions graphs were pruned only if they

conicted with H If such an edge exists in G then either there exists a

graph in the minor set of l with that edge identied and the graph has

the same edges as the graph under consideration or that graph conicted

with H Either way the merger is unnecessary

a a,b b c c

Figure Edge discrimination

Figure gives an illustration where edge discrimination would avoid joining the

two graphs b ecause there is a p otential edge illustrated by the dashed edge b etween

the no des history sets

Maximum Clique

Since graphs in a minor set can b e identied to merge with another minor set graph

then the number of no des in a minor set graph is at most the number of nodes of H

plus the linking no des of the minor set graph Thus one can build minor set graphs

where the numb er of no des exceeds the number of nodes of H In this particular

case the minor set graph has to merge with other minor set graphs that will make

that graph identify down to the number of nodes of H The maximum clique pruning

technique is based on this fact For a particular minor set graph G where the no des

of G are greater than the no des of H the maximum clique pro cedure builds a conict

graph b etween the linking no des of G where if x and y are two linking no des of G

l l

and there is a conict b etween x and y identifying together then the edge xy is in the

conict graph Some conicts include actual edges of G conicts in making a graph

that is not simple and by the edges in G After building the conict graph the

pro cedure nds the maximum clique in that graph and if the number of nonlinking

no des of G plus the maximum clique in the conict graph is greater than the number

of no des of H then G is pruned out of the minor set of l Theorem will prove

that this pro cedure will not prune away a graph isomorphic to H

Theorem Maximum clique pruning technique will not prune away

a minor isomorphic to H

Pro of From previous assumptions graphs were pruned only if they

conicted with H For the conicts by edges in G either there is a minor

set graph for l where the p ossible edges of G are identied or that graph

conicted with H Therefore the linking no des of G canonlyidentify

down to the maximum clique of G s conict graph Thus if the number

of nonlinking no des of G plus the maximum clique of G s conict graph

l l

are greater than the number of nodes of H then G will never contribute

building a graph isomorphic to H

to D

Figure oers an example of the maximum clique pruning technique where the

illustrated minor set graph has maximum clique numb er of in the conict graph

of the graphs illustrated by the edges of the graph solid edges and other conict

edges dashed edges

Figure The maximum clique in the

conict graph of this minor set graph is

Computational Results

One graph invariantthat was used to prune graphs is girth The girth ofagraph

is the smallest cycle in a graph and if the graph is acyclic then the girth is innity

So if a minor set graph has girth smaller than the girth of H then that graph can

b e pruned Girth only helps if the girth of H is ab ove All computations were

done on a MHz Pentium Pro machine with megabytes of memory and the

implementation of the algorithm was done in the C language Figure is an

example of one of the test instances where the algorithm tested to see if the Petersen

graph is a minor of tele ObviouslythePetersen graph is not a minor of tele

b ecause the Petersen graph is nonplanar and tele is planar but this test instance

was used to calculate the amount of time the algorithm would work on graphs with

ahundred or more no des while traversing through the whole tree b efore completion

As seen from Tables and the girth pruning was vital in decreasing the time for

the tele test instance mostly b ecause the girth of the Petersen graph is Girth

was also vital for most of the test instances where the minor had girth greater than

three

Figure Do es tele contain the Petersen graph P as a minor

Figure The toroidal grid C x C

For clarication of Tables and M M and Q are the graphs lab eled

M M and Q resp ectively in Figure Also C x C is a toroidal grid whichis

illustrated by Figure and P represents the Petersen graph which is illustrated

in Figure Other test instances are graphs from test instances in Chapter The

column lab eled basic is the algorithm that used only the minor graphs connectivity

the minimum and maximum degree of the minor graph and the assumption that minor

graph was simple to prune Notice that the algorithm is incorp orating more pruning

technique as one moves to the rightofthetable For example the times under MXC

are the times where the algorithm used maximum clique MXC edge discrimination

ED girth and the basic pruning techniques for minor containment testing

Table Minor ContainmentResults

graphs minor basic girth ED MXC contains minor

es colbur K y

colbur K no

colbur M no

colbur Q no

decomp K yes

decomp K no

decomp M no

decomp Q no

eld K yes

eld M no

eld Q no

fmtgen K yes

fmtgen M no

fmtgen Q no

svd K yes

svd M no

svd Q no

From Tables and one can also dra w some conclusion ab out some of the

nonplanar test instances For example a branch decomp osition of width equal to

for colbur was given in Chapter the branchwidth of colbur is at most three

From Table colbur contains the complete bipartite graph K as a minor so the

Table Minor ContainmentResults

graphs minor basic girth ED MXC contains minor

trans K yes

trans K no

trans M yes

trans M no

trans Q no

rkfs K yes

rkfs M yes

rkfs Q yes

rkfs P yes

tele P no

wldrv K yes t

twldrv K yes

twldrv M yes

twldrv Q yes

twldrv P no

C x C K yes

C x C M yes

C x C Q yes

C x C P no

us three is the branchwidth of colbur Also branchwidth of colbur is at least three Th

from Table one can see that the branchwidth of decomp is equal to three b ecause

decomp contains K as a minor but do es not contain any minor in the obstruction

set of graph with branchwidth at most three Other conclusions that can b e drawn

from Table are that the branchwidth of eld is four the branchwidth of fmtgen is

four and the branchwidth of svd is four From Table one can conclude that the

branchwidth of rkfs is ve b ecause a branch decomp osition of width for rkfs was

given in Chapter and rkfs contains the Petersen graph as minor in Section it

was shown that the Petersen graph is a member of the obstruction set of the class of

graphs with branchwidth at most The graph rkfs is illustrated in Figure Thus

one can determine the branchwidth of a nonplanar graph byknowing the obstruction

sets of all the classes of graphs formed byhaving b ounded branchwidth and the minor

containment algorithm

Figure rkfs has branchwidth

Chapter

Parallel Branch Decomp ositions

Finally since the heuristics for branch decomp ositions t nicely in a parallel scheme

this chapter describ es how to parallelize some of the heuristic co des for computing

branch decomp ositions of general graphs A sharedmemory parallel program was

implemented to compute branch decomp ositions b ecause the only ob jects that are

shared are the original graph and the branch decomp osition tree To implement

the two heuristics in parallel a multithreaded library called POSIX threads was

utilized on a machine with four pro cessors

POSIX threads or pthreads were used b ecause it is a standard library designed

to work with C or C programming Recall from Chapter the tree building

techniques for the construction of branch decomp ositions for general graphs The way

pthreads were used for the diameter and hybrid metho d was that the main thread

would nd the initial split During the pro cess threads were only created to handle

graphs that can not b e reduced by greedy separations Basically the tree building

comp onent of the heuristics was parallelized For example assume the algorithm is

working on a particular nonleaf no de v on some thread t Supp ose that the algorithm

has found a separation for v Now t will nd all the greedy separations created bythe

initial separations If there exists two new nonleaf no des whose degree is greater than

and the two nonleaf no des were created by the separation asso ciated with v thent

will check to see if the numb er of threads running is equal to the numb er of pro cessors

on the machine If this is true then t will handle b oth cases Else t will create another

thread t to handle one of those nonleaf no des Even though the pthreads library

gives the versatilitytohave more threads than pro cessors eventually one would would

lose time b ecause the threads would b e ghting for the same resources

To test for sp eedup from going from a synchronouscodetoamultipro cessor

co de the synchronous co de for the diameter and hybrid metho d was compared to a

multiprocessor co de for the diameter and hybrid metho d on a megahertz Pentium

Pro with one gigabyte of memory and four pro cessors The results of the comparison

are detailed in Tables and As one can see from the tables the parallel

co de was sometimes slower than the synchronous co de on the smaller graphs One

reason to explain this phenomena is the cost of overhead to create threads For larger

graphs the sp eedup was apparent but not signicant The maximum sp eedup from

the synchronous co de to the parallel co de using the diameter metho d was a factor of

and maximum sp eedup for the hybrid metho d was a factor of One reason

for these gures is the fact that the tree building asp ect of the heuristics was the only

comp onent parallelized One can see that if the algorithm really had a few big graphs

that needed to b e separated by the heuristic but all other graphs were separated using

greedy separations then the co de would act synchronous even though it uses multiple

threads Future work could b e conducted into actually parallelizing the heuristics

rather than the tree building comp onent of the co des This will probably oer more sp eedup

Table Parallel implementation for planar graphs

diameter hybrid

graphs branchwidth width one sec four sec width one sec four sec

atspdel

biertspdel

chtspdel

dtspdel

eiltspdel

tspdel

giltspdel

kroAtspdel

kroBtspdel

kroCtspdel

kroEtspdel

lintspdel

nrwtspdel

ptspdel

p cbtspdel

prtspdel

Table Parallel implementation for more planar graphs

diameter hybrid

graphs branchwidth width one sec four sec width one sec four sec

prtspdel

rattspdel

rdtspdel

rltspdel

tsptspdel

utspdel

vmtspdel

tele

Chapter

Conclusions and Future Work

In this work branch decomp ositions and branchwidth are dened and some insight

is given on how branch decomp ositions t into the pro of of Wagners conjecture

The rest of the thesis provide new heuristics to nd nearoptimal branch decom

p ositions for general graphs a new p olynomial time algorithm to compute optimal

branch decomp ositions for planar graphs the use of branch decomp ositions to solve

NPcomplete problems likeminimum llin and minor containment and a parallel

implementation of the new heuristics for general graphs using pthreads

For general graphs the diameter metho d and the hybrid metho d are comp etitive

to the eigenvector metho d oered by Co ok and Seymour For twenty graphs of

the test instances lab eled as planar graphs the diameter metho d found a branch

decomp osition whose width was lower than a branch decomp osition oered bythe

eigenvector metho d and the hybrid found a branch decomp osition whose width was

lower than a branch decomp osition oered by the eigenvector metho d for twenty

two graphs On the other test instances b oth metho ds found the optimal branch

decomp osition for most of the planar graphs and b oth metho ds were comparable to

the eigenvector metho d for nonplanar graphs Some future work could b e conducted

in more ways to nd separations in the graph like clique separators and an

algorithm of Kloks

For planar graphs the cycle metho d is an improvement to the edge contraction

metho d Even though the worst case analysis of the metho ds are the same the

cycle metho d found optimal branch decomp ositions faster than the edge contraction

metho d by factors ranging from to In fact the planar test instances for the

comparison had to b e shortened b ecause the edge contraction metho d would takea

considerable amount of time computing on those instances In the spirit of Ford and

Fulkerson there may exist a way to compute an optimal branch decomp osition

as one is testing for the existence of antipo dalities b ecause there seems to b e some

relationship b etween antip o dalities in the medial graph and tangles in the original

graph This task and the construction of a pro of that a u v shortest path works for

the cycle metho d if and only if any u v shortest path will work for the cycle metho d

are some directions for future work in this area The later task would prove that the

cycle metho d do es not have to rely on the edge contraction metho d to prove that the

cycle metho d will compute optimal branch decomp ositions in p olynomial time

For the minimum llin problem the heuristic is very comp etitive to the commonly

used metho ds for the problem This heuristic is simple and raw as compared to the

ne tuned and widely used metho ds in the software METIS and the heuristic

still found lower llin than those metho ds on most of the test instances Future work

in this area could include the use of a branch decomp osition to optimally solvethe

minimum llin problem In addition most minimum llin problems are mo deled

on regular and also planar graphs Esp ecially for planar graphs this heuristic may

indeed nd the optimal solution to the problem

For minor containment even though Rob ertson and Seymour gave the blueprint

for the minor containment algorithm they did not give the details of the construction

of the algorithm This algorithm is useful as a theoretical and practical to ol One

practical use was illustrated in Section were the branchwidth problem was solved

for nonplanar graphs using the minor containment algorithm An existing branch

decomp osition was used for an upp er b ound and the minor containment algorithm

tested for graphs in obstruction sets to nd a lower b ound Also there are a number of

theoretical uses for minor containment Thomas gives a survey of recent excluded

minor theorems where the minor containment algorithm could have b een used to test

some of the theorems or conjectures Some other recent excluded minor pap ers are

Maharry and Thomassen Minor containmentmay also b e used to solvethe

Jones p olynomial for knot theory

For the parallel implementation of the diameter and the hybrid metho ds not much

of a sp eedup was oered b ecause the tree building asp ect of the co des was the only

comp onentthat was parallelized The maximum sp eedup from the synchronous co de

to the parallel co de using the diameter metho d was a factor of and maximum

sp eedup for the hybrid metho d was a factor of One could p ossibly see more

of a signicant sp eedup if one parallelized the separation nding comp onentof the

heuristics instead of only parallelizing the tree building comp onent of the heuristics

Branchwidth op ens a number of opp ortunities for future work One particular

example is the use of an embedding of a graph in a orientable surface to nd an

optimal branch decomp osition of that graph Recall in Chapter an embedding of

a planar graph was used to derive a medial graph of the graph One could derivea

medial graph of a toroidal or double toroidal graph if one had an embedding of the

graph in the torus or double torus resp ectively So could one nd an p olynomial time

algorithm to nd the branchwidth and optimal branch decomp osition of a toroidal

graph using a medial graph derived from the embedding of the graph in the torus

Rob ertson and Seymour did nd a tangle not necessarily in most cases the largest

tangle for general graphs from their embedding in any surface but the answer to the

p osed question has not b een determined

The complexity of nding the treewidth of a planar graph is still an op en problem

Seymour and Thomas givea characterization of treewidth by a search game where

anumber of cops try to capture a robb er that is seen by the cops but has innite

sp eed The robb er only stands at vertices and can run at great sp eed to other vertices

along a path in the graph where no cop is standing on a vertex If there are n cops

then each of the cops are lo cated on a vertex or in a helicopter The ob jective for

the cops is to land a cop via helicopter on the vertex o ccupied by the robb er The

ob jective for the robb er is to obviously avoid capture Also the robb er can see the

helicopter approaching its designated landing sp ot and may run to a new lo cation

b efore the helicopter lands Seymour and Thomas showed that if the input graph has

treewidth at most k then k cops can capture the robb er on the graph Finding the

lowest numb er of cops such that the cops capture the robb er in the game maybe easier

if the graph is planar Also treewidth has b een dened for directed graphs This

op ens a plethora of op en problems in directed treewidth and maybe even directed

branchwidth

Some other interesting NPcomplete problems that could b e addressed using a

branch decomp osition are the chromatic numb er problem and the disjointpaths

problem The chromatic numb er problem is to nd the minimum numb er of colors to

color the no des of the graphs suchthat every no de is not adjacent to a no de with the

same color As describ ed in Chapter and Thorup this algorithm could b e used

for register allo cation for programming languages The disjoint paths problem is to

nd vertex disjoint paths b etween pairs of source and sink no des in the graph Clearly

the disjoint paths problem is a subproblem of the minor containment problem An

algorithm using a branch decomp osition can b e derived whichwould b e similar to the

algorithm describ ed byScheer

Finally another p ossible task is to nd the obstruction set for the class of graphs

with branchwidth at most Even though the obstruction sets for graphs with branch

width at most and are known the obstruction set for graphs with branchwidth at

most is still an op en problem In Section it was proved that the Petersen graph

is a member of this obstruction set but one would liketo nd a characterization of

the set and all of its members Comparably Sanders found members of the

obstruction set of graphs with treewidth at most four but did not nd all of the mem

b ers One could try to nd reduction rules for graphs with branchwidth at most four

This is similar to the work of Sanders where the author found reduction rules for

graphs with treewidth at most four Other background material that could b e used

for this problem are Arnborg Proskurowski and Corneil and Satyanarayana and

Tung

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