Group-Theoretic Algorithms and Graph Isomorphism

334 Pages·1982·8.883 MB·English

by  Christoph M. Hoffmann (eds.)

Lecture Notes ni
Computer Science
Edited yb .G Goos dna .J Hartmanis
631
IIIIII IIIIIIIll
hpotsirhC .M nnamffoH
citeroehT-puorG smhtiroglA
dna Graph msihpromosI
IIII I
galreV-regnirpS
Berlin Heidelberg kroYweN 1982
Editorial Board
W. Brauer P. Brinch Hansen D. Gries C. Moler G. Seegm~ller
j. Stoer N. Wirth
Author
Christoph M. Hoffmann
Purdue University, Dept. of Computer Science
West Lafayette, !N 47907, USA
OR Subject Classifications (t979): 3.t5, 5.7, 5.9, 5.32
JSBN 3-540-t1493-9 Springer-Vedag Berlin Heidelberg New York
tSBN 0-387-1t493-9 Springer-Verlag New York Heidelberg Berlin
rights All copyright. to subject is work This era ,devreser part or whole the whether of the lairetam
those specifically concerned, is of of illustrations, re-use reprinting, translation, ,gnitsacdaorb
or similar machine photocopying by reproduction ,snaem dna data in storage ,sknab Under
§ 54 of Copyright German the waL copies where era edam for is fee a use, private than other
elbayap Wort to "Verwertungsgesellschaft ,° Munich.
Berlin © Springer-Verlag Heidelberg by 2891
in Printed ynamreG
Printing dna Offsetdruck, binding: Beltz .rtsgreB/hcabsmeH
012345-0413/5412
This monograph develops the recent algebraic approach to Graph Isomorphism
and some of its implications for Computational Complexity. Graph Isomorphism can
be rephrased as a purely algebraic problem that exposes a surprising structural simi-
larity with a number of problems in Group Theory. These problems are easily shown
to be in NP but are not likely NP-complete. Moreover, there is a good possibility that
they are harder than Graph Isomorphism, with respect to polynomial time reduction.
Because of this possibility, the algebraic approach detailed in this book could prove to
be very important for Computational Complexity.
The roots of this approach predate Babai's Colored Graph Automorphism Problem
and my investigation of cone graphs. Nevertheless, these two papers appear to have
been the stimulus leading to the break-through subexponentia! isomorphism test for
trivalent graphs by Furst, Hopcroft and Luks. That paper already contained many of
the techniques applied later by Luks in his polynomial time isomorphism test for
graphs of fixed valence, most notably the inductive approach to determining
automorphisms. Luks' contributions have been primarily a novel way for exploiting
the imprimitivity structure of certain permutation groups and his analysis of the
structure of the automorphism groups of graphs of fixed valence.
I give my thanks to Juris Hartmanis for suggesting that this material be brought
together into a systematic survey of the area as it is at present. John Hopcroft's ded-
ication to Computer Science has been exemplary. I wish to thank him for his willing-
ness to introduce me to Graph Isomorphism. Charles Sims has been my tutor in the
mathematical aspects of this work and has been one of those rare individuals willing
to carefully read the manuscript and make suggestions for improvement. Paul Young
has been exceptionally willing to listen to my ideas and patient enough to criticize
them. Francine Berman contributed by partially relieving my teaching load. Merrick
Furst and Michael O'Donnell have thoroughly read the manuscript and improved it. I
wish to thank them all.
It is a pleasure to acknowledge the support of the National Science Foundation
(Grant Nr. SCM 78-01B12) which furthered this work. Moreover, the text processing
facilities of the Department of Computer Sciences at Purdue University have been
crucial for a timely completion.
CONTENTS
Chapter I: Introduction 1
,1 Graph Isomorphism 2
2. Computational Complexity 4
3. Group-Theoretic Algorithms 7
4. Background 9
5. Notes and References i0
Chapter lh Basic Concepts 12
.1 Review of Elementary Group Theory 12
L1. Subgroups, Cosets, Lagrange's Theorem 14
1.2. Normal Subgroups, Homomorphism, Isomorphism, and Automor-
phism 51
1.3. Permutation Groups 61
1.4. Generators, Orbits, and Stabilizers 81
1.5. Direct Products 02
2. Graph Isomorphism and Graph Automorphisms 08
2.1. Isomorphisms as Coset of the Automorphism Group 12
2.2. Some Isomorphism Complete Problems 24
2.3. Graph Isomorphism and Group Intersection 30
.3 Computationally Useful Group Descriptions 23
3.1. Determination of a Permutation Group from Generators 23
3.2. A Worked Example
14
3.3. Improvements to Algorithm 3
34
4, Accessible Subgroups 05
.5 Notes and References 85
~V
Chapter Ill: Labelled Graph Automorphisms, Cone Graphs, and p-Groups 60
,1 The Labelled Graph Automorphism Problem 60
1.1. A Deterministic Algorithm for Problem 1
.2.1 A Random Algorithm 66
2, Cone Graphs and Regular Cone Graphs 72
2,!. The Structure of the Automorphism Group of Cone Graphs of
Fixed Degree 76
.3 p-Groups and Cone Graphs 85
.3 .i Sylow p-Subgroups and Properties fo p-Groups 86
3,2. ~reath Products and Sylow p-Subgroups of n S 97
3,3, Imprimitivity of p-Groups 9t
3,4. The Central Series 100
3.5. Setwise Stabilizers in p-Groups (Method 1) 108
4. Notes and Relerences 112
Chapter IV: Isomorphism of Trivalent Graphs and of Cone Graphs of Degree
Two 114
.i The Basic Approach 115
.1 t. Properties of the Automorphism Group 115
1,2. Overall Structure of the Algorithm ii?
.3.1 Reduction to the Setwise a in Stabilizer 2-Group llg
.4.1 Binary Cone Graphs 124
,2 An Algorithm rof Determining the Automorphisms fo Trivalent Graphs 125
.3 Setwise Stabilizers ni p-Groups (Method )2 129
.3 .I The Algorithm 131
3.2. Analysis of Algorithm 2 136
4. An 0(n 4) Isomorphism Test for Trivalent Graphs 138
4, .1 Improved Algorithms for p-Groups 139
4,2, The !mprimitivity Problem for 2-Groups 157
4.3. Gadgets rof Trivalent Graph Isomorphism i67
5. Notes and References 176
IIV
Chapter V: Graphs of Fixed Valence and Cone Graphs of Fixed Degree 178
i. The Basic Algorithm i79
i.i. Outline of the Method 179
1.2. The Algorithm iSi
2. Properties of the Automorphism Group i84
3. Setwise Stabilizers in the Class b F i88
3. i. Outline of the Method 189
3.2. Group-Theoretic Preliminaries 193
3,3. The Socle of Primitive Croups 199
3.4. Primitive Croups with Nonabelian Socle 2O5
3.5. Primitive Groups with Abelian Socte 2i0
3.6. The Algorithm 2i5
4. Remarks 227
5. Notes and References 229
Chapter VI: Group-Theoretic Problems ~31
.1 Some Combinatorial Problems as Group-Theoretic Problems 231
2, Croup-Theoretic Problems of Intermediate Difficulty 235
2. i. Double Coset Problems 236
2.2. Intersection Problems 238
2.3. Miscellaneous Problems 241
2,4, Isomorphism Complete Problems 245
2.5. Remarks 245
3. A Problem with a Short Verifiable Solution 246
4. Subproblems in P 248
4.1, Group Intersection Problems 348
4.2. Centralizer and Center
4.3. An Automorphism Restriction Problem 26O
5. Normal Closure, Commutator Subgroups, Solvability, and Nflpotence 261
5.i. Normal Closure 262
5.2. Commutators and Commutator Croups 265
5.3. Testing Solvability and Nilpotence 268
6. Open Problems 27O
7. Notes and References
271
VJJl
Bibliography 273
Indices 296
,i Problem Index 296
2. Algorithm Index 298
3. Definition Index 299
4. Lemma Index 302
5. Proposition Index 305
8, Theorem Index 307
7. Corollary Index 310
NOITCUDORTNI
Two finite graphs are isomorphic if there exists a bijective map between the ver-
tex sets of the two graphs which preserves adjacency. Determining whether two given
graphs are isomorphic is a problem of both practical and theoretical interest, and
there has been extensive work investigating whether graph isomorphism can be
tested efficiently. Despite much work, there is to date no polynomial time test for
graph isomorphism, nor is there a proof that no such test can exist. Nevertheless,
there do exist algorithms which can test isomorphism of certain classes 0f graphs in
polynomial time.
The two dominant lines of attack on graph isomorphism are topological and
group-theoretic. In this monograph, we give a comprehensive development of the
group-theoretic approach. Within a very short time, this approach has substantially
broadened the class of graphs for which there exist polynomial time isomorphism
tests and it has stimulated interest in a number of algebraic problems which had not
been previously investigated for their computational complexity.
In the group-theoretic approach, one determines the group of all automorphisms
of the graph. It is not too difficult to prove that knowledge of the automorphism
group enables one to also test for isomorphism. At the same time, the translation of
the topological question of testing graph isomorphism into the algebraic question of
determining the symmetries of the graph allows the use of many new computational
techniques. In particular, there are many algorithms for determining properties of
permutation groups which become applicable for tests of graph isomorphism. The
group-theoretic approach enables one to view the problem of graph isomorphism
from a new perspective. Because of this, the algebraic approach has rapidly yielded
significant new results.
Most of the algorithms developed in this monograph have been selected because
of their bearing on the graph isomorphism problem. However, graph isomorphism
can also be generalized, yielding a spectrum of algebraic problems ~o apparently
greater difficulty. The generalizations are natural in the sense that they have the
same "structure" as graph isomorphism, but they are also generalizations in the
computational sense that a polynomial time al~orithm rof the more general problem
implies a polynomial time algorithm rof graph isomorphism. We investigate those
problems as .llew
The generalizations algebraic fo graph isomorphism reffid from previous generali-
at in zations least two tnacifingis aspects: ,tsriF they are not themselves isomorphism
questions, and second, while these problems clearly are ni NP, there si as (just ni the
case fo graph isomorphism) some technical evidence that the generalizations are not
NP-eomplete. Thus we conjecture that these generalizations are part fo a hierarchy
fo increasingly more tluciffid problems within NP.
i. Graph Isomorphism
Graph isomorphism has received considerable attention rof a number fo reasons,
both practical and theoretical. For example, ni combinatorial studies, one frequently
wishes to generate a tsil fo combinatorial objects ni which each object occurs exactly
once. This can be done ni two phases: ,tsriF a tsil si constructed ni which each object
appears once. least at Then, multiple occurrences are deleted. Usually, these objects
correspond in some natural sense to graphs, so recognizing multiple occurrence fo an
requires testing object isomorphism fo graphs. Aost her applications fo graph isomor-
phism, combinatorial designs, scene analysis, and chemical documentation are fre-
quently .detic
Part fo the theoretical appeal fo graph isomorphism si sti unknown complexity
.sutats si tI a clearly problem ni NP, si ti not known to be in ,P and theraer e proper-
seit fo the problem which seem to make ti unlikely ot be NP-complete. Thus, there si
a good ytilibissop that si ti a problem fo intermediate ,ytluciffid ,.e.i a problem which
si neither ni P nor si NP-complete.
As mentioned above, the two dominant approaches ot graph isomorphism are the
topological approach and the group-theoretic approach. nI the topological approach,
one embeds the graphs onto surYace a fo minimal genus. Then one dissects the sur-
face (and with ti the graph) into planar components. With careful a study fo the pos-
elbis interconnection structure, one then reduces testing isomorphism fo graphs
embedded a in surface fo nonzero genus testing to isomorphism fo planar graphs.
The topological approach leads to the ge~s A~eT~TcAy: For each genus ,g there
stsixe a polynomial pg degorfe e f(g) such that isomorphism fo graphs embeddable
onto surface a fo genus g can be tested at in most steps, pg(n) where n si the number
fo vertices fo the graph.
Note the that problem fo determining the graph genus si fo comparable .ytluciffid
That ,si the algorithm rof determining the genus fo a graph requires time proportional
a to polynomial whose degree grows with the graph genus.
In the group-theoretic approach to graph isomorphism, one seeks to determine a
tes fo generating permutations rof the automorphism group fo the graph. tI can be
shown that testing graph isomorphism si polynomially reducible to determining gen-
erators rof the automorphism group fo graphs and that there are small sets fo gen-
erating permutations. We prove this result ni Chapter .II the Typically, group-
theoretic approach si based on structural properties fo the automorphism group
which are the result fo graph properties such as the valence fo the graph. While the
graph properties exploited are often ,laivirt the group-theoretic resulting properties
may be quite complex.
As an noitartsulli fo the group-theoretic approach, consider determining the
automorphism group fo a connected trivalent graph .X Here one proves tshueb -t hat
group consisting fo those automorphJsms which xif an arbitrarily chosen edge ni the
graph si g-group; a that ,si si ti a group each fo whose elements has order a power fo
.2 tI turns out that 2-groups have special properties which can be in exploited the
design fo algorithms to determine the intersection fo two groups, one fo which si a -2
group. These algorithms run ni time polyno~al in the number fo graph ,secitrev
despite the fact that the order fo these groups might very well be exponential. Such
an algorithm serves as the basis rof an isomorphism test fo trivalent graphs. We
develop this approach ni Chapter .VI
The group-fiheoretic approach leads to the ~aLe~ce hierarchy: For each valence ,d
there a exists polynomial Pd fo degree h(d) such that isomorphism fo a graph X fo
valence d can be tested ni steps, pd(n) where n si the number fo vertices .X fo This
result si presented ni Chapter .V si tI to possible broaden ehif class fo graphs handled
by this approach ni polynomial time, and ni Chapters IV and V we discuss which
graph-theoretic properties are required.