Algorithms for Random Generation and Counting: A Markov Chain Approach

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Progress in Theoretical Computer Science

Alistair Sinclair Editor Ronald V. Book, University of Californa

Editorial Board Erwin Engeler, ETH Zentrm, Zurich, Switzerland Gérard Huet, INRI, Le Chesnay, France

Jean-Pierre Jouannaud, Université de Pars-Sud, Orsay, France Robin Milner, University of Edinburgh, Edinburgh, Scotland

Algorithms for Random Generation and Counting: A Markov Chain Approach

Maurice Nivat, Université de Pars VII, Pars, France

Marin Wirsing, Universität Passau, Passau, Germany

Birkhäuser Boston' Basel, Berli

Contents Alstar Sinclair

Deparent of Computer Science

University of Edinburgh Edinburgh, Scotland

Synopsis Library or Congress Cataloging-in-Publication Data Sinclair, Alista, 1960 Algorithms for random generation and counting : a Markov chain approach I Alistar Sinclair. p. em. -- (Progress in theoretical computer science)

Includes bibliographical references and index.

ISBN 0-8176-3658-7 (acid-fre) 1. Markov proesses. 2. Combinatorial set theory. 3. Algorithms. I. Title. II. Series.

QA274.7.S56 1992

519.2'33--dc20

92-34616 CIP

1 Preliminaries

(g Birkhäuser Boston 1993.

Portions of Chapters 1, 2 and 4 are reprinted with permission from Informtion and Computation, vol. 82, pp. 93-133. (g 1989 Academic Press. Portions of Chapter 3 are reprinted with permission from the SIAM Journl on

Computing, vol. 18, no. 6, pp 1149-1178. (g 1989 by the Society for Industral and Applied Mathematics. Copyright is not claimed for works of U.S. Government employees. All rights reserved. No par of this publication may be reproduced, stored in a retreval system, or trsmitted, in any form or by any means, electrnic, mechanical, photo-

copying, reording, or otherwise, without prior permssion of the copyright owner.

Permssion to photocopy for internal or personal use of speific clients is grante by Birkhäuser Boston for libraes and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $6.00 per copy, plus $0.20 per page is paid ditly to CCC, 21 Congress Stret, Salem, MA 01970, U.S.A. Speial requests should

7

1.1 Some basic definitions

7

1.2 Notions of tractabilty

12

1.3 An extended model. .

18

1.4 Counting, generation and

self-reducibilty . . . . . . 1.5 An interesting class of relations

2 Markov chains and rapid mixing Prte on acid-fre paper.

1

28

36

42

2.1 The Markov chain approach to

generation problems . . . . . 2.2 Conductance and the rate of

43

convergence . . . . . . . . . .

46

2.3 A characterisation of rapid mixng

56

3 Direct Applications

63

3.1 Some simple examples . . . . .

63

3.2 Approximating the permanent

70

3.3 Monomer-dimer systems

85

3.4 Concluding remarks

96

be addressed directly to Birkhäuser Boston, 675 Massachusetts Avenue, Cambridge, MA 02139, U.S.A.

ISBN 0-8176-3658-7 ISBN 3-7643-3658-7 Camera-ready copy prepared by the Author in TeX. Prnte and bound by Quinn-Woobine, Woobine, NJ. Prte in the U.S.A.

987654321

4 Indirect Applications

101

4.1 A robust notion of approximate counting . . . . . . . . . .

102

4.2 Self-embeddable relations

112

4.3 Graphs with specifed degrees

114

~

Appendix: Recent developments

126

Bibliography

136

Index

144

Foreword This monograph is a slightly revised version of my PhD thesis (86), completed in the Department of Computer Science at the University of Edinburgh in June 1988, with an additional chapter summarising more recent

developments. Some of the material has appeared in the form of papers (50,88).

The underlying theme of the monograph is the study of two classical problems: counting the elements of a finite set of combinatorial structures, and generating them uniformly at random. In their exact form, these prob-

lems appear to be intractable for many important structures, so interest has focused on finding effcient randomised algorithms that solve them approxim~ly, with a small probabilty of error. For most natural structures the two problems are intimately connected at this level of approximation,

so it is natural to study them together. At the heart of the monograph is a single algorithmic paradigm: sim-

ulate a Markov chain whose states are combinatorial structures and which converges to a known probabilty distribution over them. This technique has applications not only in combinatorial counting and generation, but also in several other areas such as statistical physics and combinatorial optimisation. The effciency of the technique in any application depends crucially on the rate of convergence of the Markov chain. Since the number of states

reach eqlÙlibrium after exploring only a tiny fraction of its state space; chains with this property is typically extremely large, the chain should

are caled rapidly mixing. A major portion of the monograph is devoted

to developing new methods for analysing the rate of convergence, which are of independent mathematical interest. The methods are quite general and lay the foundations for the analysis of the time-dependent behaviour of complex Markov chains arising in the above applications. The power of these methods is ilustrated by using them to analyse several interesting Markov chains. As a result, polynomial time approximation algorithms are obtained for counting and generation problems associated

with various natural structures, including matchings in a graph and labelled graphs with specified vertex degrees. Important consequences are the first known polynomial time approximation algorithms for computing the permanent of dense 0-1 matrices and the partition fuction of monomer-dimer systems in statistical physics. Markov chain techniques are also used in the monograph to provide futher insight into the relationship between counting and generation problems in general, and to establish a pleasing robustness result for a suitable notion of annrovim:itp rmmtinO"

"

Since the thesis appeared in the Summer of 1988, there has been much interest in the quantitative analysis of Markov chains and their computational applications. The analytical methods developed here have been refined and applied to many important new examples, and the subject is rapidly becoming a research area in its own right. Were I to embark on writing a new monograph from scratch today, I would take as the starting point the notion of a rapidly mixng Markov chain and proceed to the

Synopsis This monograph is concerned with two classes of problems involving a finite set S of combinatorial structures: counting them and generating them at

random from a uniform distribution. The set S wil always have a concise

applications from there. In recogntion of these developments, I have at-

description x: for example, x may be a connected graph and S its set

tempted to summarise in an Appendix the most signcant activity in this

of spanng trees, or x a positive integer and S its set of partitions. We

field up to early 1992. References are given to the relevat papers so that the interested reader may follow up the details. I hope that the monograph

also consider the more general setting in which each structure in S has

wil provide a usefu introduction to what promises to be an exciting and

sum of all elements of S, or to generate them randomly with probabilties

fruitful research area.

proportional to their weights.

I wish to record my warmest thanks to Mark Jerrum, my PhD supervisor, for numerous suggestions and discussions during the development of this work and for reading everything so criticaly. I would also like to thank Clemens Lautemann, Kyriakos Kalorkoti and Gordon Brebner for helpfu discussions, and my friends and colleagues at the Deparment of Computer Science at Edinburgh for providing a pleasant and stimulating research environment.

which we wil not attempt to summarise here. In addition to their in-

an associated positive weight: the task is then to compute the weighted

Combinatorial counting problems have a long and distinguished history

trinsic interest, they arise naturally from investigations in numerous other branches of mathematics and the natural sciences and have given rise to a rich and beautifu theory.

Generation problems are less well studied but have a number of computational applications. For example, uniform generation can be seen as a way of exploring a large set of combinatorial structures and constructing

typical representatives of it. These may be used to formulate conjectures about the set, or perhaps as test data for the empirical analysis of some

heuristic algorithm which takes inputs from the set. Non-uniform genera-

Rutgers University, January 1992

tion occurs in the mathematical modellng of physical systems where the structures are vad system confguations each of which has a weight which

depends on its energy. Many important properties of the model can be deduced from estimates of the expectation, under the weighted distribution, of certain operators on confguations. An analogous idea lies at the hear

of widely used stochastic techniques for combinatorial optimisation, such as simulated annealng. Here, low cost solutions are assigned low "energy" and thus large weight: sampling from the weighted distribution therefore tends to favour them.

The central question we shal address is that of the existence of com-

putationally effcient counting and generation procedures for a given class of structures. In accordance with accepted practice in theoretical computer science, "effcient" is taken to

mean having a runtime which grows

only polynomialy with the size of the problem description x. Of course, 1

I

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SYNopsis

2

i-,

merating the elements of S since there wil in general be far too many of them. Our treatment is motivated by the empirical observation that effcient teresting structures. For many others, the apparent hardness of counting

is supported by strong theoretical evidence to the effect that no effcient counting procedure exists unless P = NP. However, there is no a priori

3

suggest themselves for one class of problems can be used to solve the other indirectly.

for most structures this rules out the naïve approach of exhaustively enu-

exact counting procedures exist only for a relatively small number of in-

SYNOPSiS

Traditionaly the cross-fertilzation between these problems has tended F

to be rather one-sided: exact or approximate analytic results on count-

ing enable one to generate structures uniformly or alost uniformly. This monograph concentrates chiefly on the opposite direction. We analyse a powerfu algorithmic paradigm for generation, which by virtue of selfreducibilty constitutes an attack on counting problems as well.

reason to suppose that some of these structures cannot be counted ap-

The paradigm in question, which has been in use for some years in

proximately in some suitable sense. Specificaly, it is interesting to relax the requirements of exact counting in two directions: randomisation and approximation. Thus we alow our counting procedures to flip coins, and

statistical physics under the name of the Monte Carlo method (12), involves

demand that they produce an answer which is correct to within some small

to move around the state space by means of random local perturbations

specified error with high probabilty. In the case of generation, we settle for

of the structures. Moreover, the process is ergodic, i.e., if it is alowed to

a distribution which is almost uniform (or, more generally, close to some

evolve in time the distribution of the final state tends asymptoticaly to a unque stationary distribution 7r over S which is independent of the initial

weighted distribution) with some small specified bias. This gives us well-

constructing a dynamic stochastic process (a finite Markov chain) whose states correspond to the set S of structures of interest. The process is able

defined notions of approximate counting and almost uniform generation: a

state. By simulating the process for sufciently many steps and outputting

fundamental assumption implicit in our approach is that they correspond to reasonable notions of effective tractabilty.

the final state, we are therefore able to generate elements of S from a

The principal aim of this monograph is to go some way towards classifying naturally occurring counting and generation problems according to these notions of tractabilty. We shall concentrate mainly on techniques for

proving positive results, though it is worth pointing out that for certain structures the problems remain hard even in approximate form. Our investigation wil have essentialy two strands: the classifcation of particular structures, and the nature of the classification process itself.

The reader may be wondering why we have chosen to investigate two apparently very disparate classes of problems at the same time. The reason

for this is that, from a computational point of view, the problems of approximate counting and almost uniform generation are very closely related. More precisely, for most natural structures a polynomial time procedure for approximate counting can be used to construct a polynomial time almost uniform generation algorithm, and vice versa. The only assumption we need to make is that the structures are self-reducible, which essentialy

means that they possess a simple inductive construction in terms of similar structures of a smaler size. This connection has frequently been observed

in specific cases, and was formalised in a general setting by Jerrum, Valiant and Vazirani (53), whose work initially inspired the author to embark on

the research presented 'here. The upshot is that counting and generation problems may profitably be studied in tandem: techniques which naturally

distribution which is arbitrarily close to 7r. The construction and simulation of an ergodic Markov chain with the

desired stationary distribution is usualy straightforward. What is not at al obvious, however, is how to determine the number of simulation steps which are necessary in order to achieve some specifed degree of accuracy.

In most application areas, such as Monte Carlo simulations in physics, ad hoc arguments are typically used to estimate this number. There is clearly a need for rigorous analytic bounds if we are to have confdence in the results produced by such methods.

According to our effciency criterion, the crucial consideration is that the number of steps required for the chain to be close to stationarity should grow only polynomially with the size ofthe problem description x. We refer

to chains which have this property as rapidly mixing. Since the number of states wil in general be exponentialy large, rapid mixng is a strong property: it demands that the process should be close to equilibrium after visiting only a tiny fraction of its state space.

The classical theory of Markov chains is concerned alost exclusively with asymptotic behaviour and provides little useful information about the rate of convergence. More recently, several authors have proposed various methods of analysis but these have so far been able to handle only sim-

ple chains (such as random walks on cubes) which possess a highly symmetrical structure. The first major contribution of this monograph is to

!n r L

SYNopsis

4

f,

l

establish a useful characterisation of the rapid mixng property for a wide

5

of certain stochastic optimisation heuristics such as simulated annealing. Indeed, some substantial progress in these directions has aleady been made

class of Markov chains. The characterisation is based on a structural prop-

erty, called the conductance, of the weighted graph underlying the chain. The conductance is a global measure of connectedness, or communication strength, of the process and is a natural analogue in this context of the more familar graph-theoretic concept of expansion. Informaly, our characterisation says that a (reversible) Markov chain is rapidly mixng if and only if the conductance is large. This result represents a generalisation of recent work by Alon and Milman (4, 5) on the connection between the eigenvaues

SYNOPSiS

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and is reported in the Appendi. The final theme of the monograph is concerned more with the general notions of approximate counting and alost uniform generation than with paricular structures. For approximate counting, we are able to establish

i

certain Markov chains has been observed by Aldous (2) and others.

a pleasing robustness result: for any self-reducible structures, approximate counting within a factor of the form 1 + nß is possible in polynomial time either for al real constants ß or for no ß. Thus we are justifed is callng a counting problem tractable if it can be aproximated within some such factor by a polynomial time randomised algorithm. This provides a natural way

More signficant than the characterisation itself is the fact that it allows the rate of convergence of non-trivial Markov chains to be analysed

of classifying hard counting problems with respect to a well-defined notion of approximabilty.

for the first time. In order to do this, we develop a general methodology for

The mechanism used in the above proof is of independent interest. The key step is showing how to generate structures alost uniformly given only very crude counting information for them. This is achieved by means of a Markov chai suggested by the self-reducibilty inherent in the problem,

of a graph and its structural properties, whose relevance to the analysis of

deriving lower bounds on the conductance of the underlying graphs. The methodology is based on constructing paths between states in the underlying graph in such a way that no edge carries too many paths. A novel injective mapping technique is introduced to bound the number of paths traversing any edge.

With this machinery in place, we show how to infer the rapid mixng property for natural Markov chains whose states are matchings (independent sets of edges) in a graph. (Chains of this kind were fist considered by Broder (16).) We are therefore able to generate alost uniormly in polynomial time the following classes of structures: perfect matchings in a very

large class of graphs, includig al sufciently dense graphs, and matchings of al sizes in arbitrar graphs. As a result, we establish for the first time the existence of effcient approximation algorithms for counting these structures. Both counting problems are highly signcant, and are known to be hard in exact form. Counting perfect matchings in a bipartite graph is equivaent to computing the permanent of a 0-1 matrix, a problem for which mathematicians have been seeking an effcient computational procedure for over a centur. Counting and generating matchings is of interest in

the context of monomer-dimer systems in statistical physics. In the latter case, we are also able to handle natural weighted versions of the problems, and so to approximate the partition function of a monomer-dimer system. We conjecture that the methods described above point the way towards

a powerful general approach for analysing the effciency of algorithms based

on finite Markov chains. Potential future application areas include fuher positive results on approximate counting, rigorous performance guarantees for Monte Carlo simulations in statistical physics, and the demystifcation

I: i, II I

which can again be analysed using the characterisation described earlier. This construction can help us to exploit a much wider range of analytic results on approximate counting than has hitherto been possible in this context, even for structures which are not self-reducible. To ilustrate this point, we consider the problem of generating almost

uniformly labelled graphs with given vertex degrees and a given excluded subgraph. Using an asymptotic result of McKay (69) on the number of such graphs, we are able to generate them in polynomial time from a distribution which is very close to unform, provided that the vertex degrees are not too

large. The range of degrees we can handle is considerably larger than for previous methods. Moreover, we show the existence of polynomial time

algorithms for approximating the number of graphs with a much smaller error than that of avaable analytic estimates.

The monograph is organsed as follows. In Chapter 1 we present some fudamental definitions and summarise earlier work which is relevat to our

exposition. The general results on Markov chains are discussed in Chapter 2, culinating in the characterisation of the rapid mixng property. The next two chapters are devoted to applications of the characterisation. In Chapter 3 we consider Markov chains for generating specific structures, and establish positive results for the permanent and monomer-dimer systems.

In Chapter 4 we discuss the robustness of our notions of approximate counting and uniform generation, and apply some of our new general machinery

6

SYNopsis

to the degree-constrained graph problem mentioned above. Finaly, in the Appendi we summarise recent developments in the theory and applications of rapidly mixng Markov chains that apply and extend the techniques presented in the monograph.

I i ¡

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Chapter i

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Preliminaries

The aims of this introductory chapter are twofold: to set down and motivate precise definitions of some fudamental concepts, and to briefly summarise relevant previous knowledge. Accordingly, much of this material is not new,

but is included in the interests of orienting the reader. f i i

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Three concepts defined here wil playa central rôle in all that follows: self-reducibilty, approximate counting and almost uniform generation. We trust that the reader wil gain an intuitive feel for them through the mostly elementar discussions of this chapter. Almost all the algorithms we shall present in this monograph are probabilstic or randomised. We therefore

devote some effort at this stage to a discussion of the model of randomised computation so that technical details may be avoided in the sequel.

The starting point for our later investigations is the work of Jerrum, Valant and Vazirani ¡53), in which the intimate computational connection between counting and generation was fist formalsed. Since this connection

underlies most of our later work, we feel it is appropriate to spell it out in

some detail in Section i.4. Finally, we set the scene for the rest of the monograph by identifying a class of combinatorial structures for which the questions of approximate counting and uniform generation are particularly relevant.

1.1 Some basic definitions We begin by introducing a framework within which a number of natural classes of combinatorial problems, including counting and generation, may be formulated and their computational complexities compared. Common

to al classes is the idea of computing certain information about a finite but usualy very large set of combinatorial structures given a concise implicit description of the set. Tyicaly, the description takes the form of some

1.1 SOME BASIC DEFINITIONS

8

other combinatorial entity drawn from a family of problem instances, together with a relation R which associates with each instance x in the family a finite set R(x) of structures called solutions, as in the following examples:

1. Problem instances: Boolean formulae B in disjunctive normal form (DNF); Solution set R(B) : al satisfying assignments of B.

2. Problem instances: undirected graphs G;

I-factors (perfect matchings) of G.

Solution set R(G) : al

3. Problem instances: positive integers n; Solution set R( n) : al partitions of n.

1) over which instances and solutions are to be encoded, and let R ç E* x E* be a binary relation over E. For any string x E E*, the corresponding solution set with respect More formally, we fi a finite alphabet E :2 to,

to R is just R(x) = tY E E*: (x,y) E Rl.

Note that no distinction is made between strings which do not encode a "valid" problem instance and those which encode a problem instance with empty solution set. The formal counterpart of example 1 above is then R = t (x, y) : x E E* encodes a Boolean formula B in DNFj y E E* encodes a satisfying assignment of B l.

Throughout this monograph we shall move freely between the formal and informal problem descriptions without explicitly saying so, assuming always that the encoding scheme used is "reasonable" in the sense of Garey and Johnson ¡38). The relational framework used here is taken from Jerrum, Valiant and Vaziran ¡53), and has alo appeared elsewhere, notably in the form of the "search fuctions" of Valiant ¡95) and the "string relations" of Garey and Johnson ¡38).

Our main concern is with classes of problems which involve computing the cardinalty of a finite set of combinatorial structures or generating el-

ements of the set unformly at random. In fact, most of what we wi say applies in the more general setting where each structure has an associated

I i

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i I I

1.1 SOME BASIC DEFINITIONS

9

Formaly, the counting problem for a relation Rover E involves com-

puting the function #R : E* -- N defined by #R(x) = IR(x)l. In the ( uniform) generation problem for R, on input x E E*, we are asked to

select an element of R(x) at random in such a way that each solution has equal a priori probabilty of being chosen. We shall say in the next section precisely what we mean by effective solutions to these problems. For the purposes of comparison, we mention some other important prob-

lem classes which fit naturally into this framework and which have been extensively studied. Foremost among these is the existence problem (or

decision problem) for a relation R, which on input x E E* asks whether the solution set R(x) is non-empty: in example 1 above, this means asking whether a given DNF formula is satisfiable. The construction problem is similar, but requires in addition that some solution y E R(x) be output if one exists. If some cost fuction is defined on solutions, then the optimi-

sation problem seeks a solution of minimum cost. Finaly, the enumeration problem for R involves listing all elements of the solution set R(x).

We shal restrict our attention throughout to relations whose solutions are easy to check, i.e., for which membership of a candidate object in a

given solution set can be tested effciently. Let Ixl denote the length of the string x. Borrowing terminology from ¡53), we say that R is a p-relation if

(prI) There exists a polynomial p such that, for al strings x, y E E*, (x, y) E R ~ Iyl $ p(lxl) .

(pr2) The predicate (x, y) E R can be tested in time polynomial in Ixl+lyl. The class of existence problems associated with p-relations may be identified with the more familar class NP of languages accepted by polynomialy time- bounded nondeterministic Thing machines (NP-machines). Specif-

caly, if M is a nondeterministic Thing machine with at most lEI possible moves from each confguation, then any halting computation of M may be encoded in a standard way as a string over E. It should be clear that R is a p-relation if and only if there exists an NP-machine M such that, under such an encoding,

R = t (x, y) : y encodes an accepting computation of M on input x l.

positive weight, and the task is to compute the weighted sum of the struc-

tures or to generate them randomly with probabilties proportional to their weights. For the sake of simplicity we shal work with unweighted problems until Chapter 3, where weighted structures wil arise naturaly in one of our

applications.

In precisely the same manner, Valant's class #P ¡96) may be viewed as the class of counting problems associated with p-relations. (A fuction

f : E* -- N belongs to #P iff there exists an NP-machine M such that, for al x E E*, the number of accepting computations of M on input x is j(x).)

10

1.1 SOME BASIC DEFINITIONS

1.1 SOME BASIC DEFINITIONS

For many naturally occurring p-relations, the structures in each solution set have a simple inductive construction from the solution sets of a few smaler instances of the same relation. All of the above examples possess

this property: in the case of DNF satisfiabilty, for instance, there is an obvious (I-I)-correspondence between the satisfying assignments of Band those of the reduced formulae BT and B F, which are obtained from B by

i I I

11

R(x) = ø or R(x) = -(Al. Note that this, together with condition (sr3), implies that we can test in time polynomial in ixl + lyl whether a candidate solution y E E* belongs to the solution set R(x). In view of condition (srI), any self-reducible relation is therefore automatically a p-relation.

its vaiables the vaues TRUE and FALSE respectively.

Example 1.1 Consider the relation MATCH which associates with an undirected graph G all matchings (independent sets of edges) of G. Then MATCH

This property is generally known as self-reducibility and was first studied

is easily seen to be self-reducible since, for an arbitrary edge e = (u, v) of G,

substituting for one of

by Schnorr ¡83J. Formaly, we say that a relation Rover E is (polynomial

MATCH(G) = MATCH(G-) U t M U -(el: M E MATCH(G+) ì,

time) self-reducible iff

(srI) There exists a polynomial time computable length function lR E* -- N such that lR(x) = O(lxlkR) for some constant kR, and

Y E R(x) :: lyl = lR(x)

Vx,y E E*.

where G- is the graph obtained by deleting the edge e from G, and G+ the graph obtained by deleting the vertices u and v together with all their

incident edges. D From our point of view, the crucial feature of the above definition is that it alows the counting or uniform generation problem for any instance x to

(sr2) For all x E E* with lR(x) = 0, the predicate A E R(x) can be tested in polynomial time. (A denotes the empty string over E.)

We shall exploit this fact later in establishing a close relationship between

(sr3) There exist polynomial time computable functions 'i : E* x E* -E* and a : E* -- N satisfying

these two problems for a given self-reducible relation. The following two properties of a self-reducible relation R are easily checked and already well

a(x) = O(lg Ix!);

lR(x) :: 0 Ç: a(x) :: 0

v x E E*;

1'i(x,w)1 :: ixl

Vx,w E E*;

lR('i(x,w)) = max-(lR(x) -lwl,Ol

Vx,wEE*,

and such that each solution set can be expressed in the form

R(x) = U t wy: y E R('i(x,w)) ì. w E EtT(x)

Condition (sr3) provides an inductive construction of the solution sets as

be formulated in terms of the same problem for smaler instances 'i( x, w).

known: if the existence problem for R lies in P, then (i) the construction problem for R can be solved in polynomial time; and (ii) the enumeration problem for R can be solved in time #R(x)p(lxl), where x is the input andp is a polynomial. In fact, these relationships follow from weaker notions of self-reducibilty, in which the strict (I-I)-correspondence of condition (sr3) is replaced by a requirement to the effect that a solution to the existence problem for any instance can be derived easily from solutions for certain subinstances. Such notions have been explored by various other authors

(see, e.g., ¡73, 84)). Despite the fact that our definition appears rather strong, we have to work quite hard to find natural relations which cannot easily be formulated so as to be self-reducible. An outstanding candidate is the relation which associates with a positive integer n the set of factors

follows: if the solution length L R (x) is greater than zero, R( x) can be parti-

ofn.

tioned into classes according to the (smal) initial segment w of length a(x),

It is conceptualy helpfu to capture the inductive construction of solutions of a self-reducible relation explicitly in a tree structure. Suppose that

and each class can then be expressed as the solution set of another instance 'i(x, w), concatenated with w. The partitioning of satisfying assignments of a DNF formula indicated above is easily seen to be of the required form (under some natural encoding). An atom is an instance x E E* with

problem instance label inst( v) E E* and a partial solution label sol( v) E E*,

solution length lR(x) = 0: in the above example, these would include (en-

defined inductively as follows:

co

the relation Rover E is self-reducible. For each x E E* with R(x) -l ø, the tree of derivations TR(x) is a rooted tree in which each vertex v bears both a

dings of) the constants TRUE and FALSE, viewed as DNF formulae. Con-

dition (sr2) says that, for atoms x, we can test in polynomial time whether

(tI) The root u has labels inst(u) = x and sol(u) = A.

1.2 NOTIONS OF TRACTABILITY

1.2 NOTIONS OF TRACTABILITY

12

a PTM is a Tuing machine with output tape (45, Chapter 7) which has

(t2) For any vertex v in TR(x), if the problem instance z = inst(v) is an

certain distinguished "coin-tossing" states. The machine is deterministic

atom then v is a leaf. Otherwise, define

W(v) = t wE EU(z) : R('l(z,w)) =f ø 1.

13

i I

except that in coin-tossing states precisely two possible transitions are spec-

ified, the choice of transition being determined by the toss of an unbiased coin. If the machine halts, its output is the contents of the output tape;

(Note that W(v) is non-empty.) Then v has a child Vw for each

otherwise, its output is undefined. A PTM is deterministic if it has no

wE W(v), with labels inst(vw) = 'l(z, w) and sol(vw) = sol(v) . w.

coin-tossing states.

Note that the labels sol(v) are distinct, whie the inst(v) are in general not. It should be clear that the labels sol( v) for leaves v are precisely the elements of R(x), so there is a (I-I)-correspondence between leaves and solutions.

More generaly, for any vertex v of TR(x) there is a (I-I)-correspondence between the solution set R (inst( v)) and the leaves of the maxmal subtree

rooted at v. The bounds on 0' and 'l in the definition of self-reducibilty

ensure that the depth of TR(x) is at most lR(x), and hence bounded by a polynomial in lxi, and that its vertex degree is alo polynomialy bounded. The definition of W(v) above makes it clear that, in order to infer the

structure of the tree of derivations, it is necessar to solve the existence problem for the relation in question. Since we wil not always be able to do

The above defition induces a probabilty distribution on the set of computations of a PTM M on a given input. We define the random vaiables M(x) and ComPM(x) to denote the output of M on input x and the computation of M on input x respectively, with the proviso that M(x) takes

the special vaue 1. when the output is undefined. We restrict attention to PTMs with alphabet E U -(?1, where? rt E is a distingushed symbol which may appear only on the output tape and is to be interpreted as rejection or

faiure. A computation of a PTM is accepting if it halts with some output other than ? According to the application at hand, vaious definitions for

the time complexity of a PTM are possible. We define the runtime of a PTM M on input x to be the length of a longest accepting computation of M on x.

this with certainty, it is usefu to define the self-reducibility tree TR(X) as for T R ( x) above except that the restriction R( 'l (z, w)) =f ø in the definition

of W(v) is removed. Clearly, TR(X) contains TR(x) as a subgraph and their labels agree. All solutions in R(x) stil occur precisely once as labels of

leaves ofTR(x), but there may be other leaves whose parial solution labels are not in R(x). The depth and vertex degree ofTR(x) remain polynomialy bounded as before.

Let us now consider solutions to the counting problem for p-relations. Our approach here is motivated by the observtion that effcient exact solutions to this problem are the exception rather than the rule. By an effcient solution we mean a deterministic algorithm which computes #R(x) in time polynomial in the input size Ixl. Note that the naïve method of explic-

itly enumerating candidate solutions is unacceptable since in general their number grows exponentially with the size of the input. Indeed, the class

1.2 Notions of tractability The aim of this section is to define precisely what is meant by an effective

solution to the counting and generation problems introduced earlier. First we must agree on a model of computation: as we wil chiefly be studying

#P of counting problems associated with p-relations is not known to lie

within any level of the Meyer-Stockieyer polynomial time hierarchy (90). More strikngly, there are many natural p-relations (among them examples 1 and 2 of the previous section) whose counting problem is complete

able model, so we shal use an elementar computing device with access

for #P, i.e., in a precise sense as hard as any problem in the class, but whose construction problem is solvable in polynomial time: thus whie finding a perfect matching in a graph is easy, counting these structures is apparently intractable. A detailed discussion and fuher examples of this phenomenon

only to a stream of truly random bits. Possible extensions of this model

can be found in (97).

probabilstic algorithms, some source of randomisation has to be included.

It is clearly desirable to formulate definitions in terms of the simplest ava-

wil be considered in the next section. Aprobabilistic Turing machin'e (PTM), as defined by Gil (39), is a stan-

dard deterministic Tuing machine equipped with the additional abilty to make decisions according to the fal of an unbiased coin. More formally,

För functional evauation problems which have resisted attempts at an

exact solution, it is natural to seek effcient approximation algorithms which estimate the vaue of the function within a specified factor. This notion is

familar from combinatorial optimisation (see, e.g., (38, Chapter 6)) and

14

1.2 NOTIONS OF TRACTABILITY

1.2 NOTIONS OF TRACTABILITY

asymptotic analysis ¡ioJ. (A less conventional, and much more severe definition of approximation is studied by Cai and Hemachandra ¡18J.) A sec-

i

ond approach towards reducing the complexity of apparently intractable

l

problems is to alow some element of randomisation in the algorithm, requiing only that it delivers a reliable solution with high probabilty. Whie rigorous proofs that randomisation helps are avaable only in rather re-

stricted circumstances, probabilstic algorithms which run faster than their best known deterministic counterpars have been discovered for a vaiety

15

(f.p.) if its runtime is bounded by a polynomial in ixl and c1. (In practice,

we can think of the vaue c1 as being specified on the input tape as an integer in unary notation.)

Ii E

I

The significance of the lower bound of 3/4 in the above definitions lies in

the fact that it allows the counters to be "powered" so that the probabilty of producing a bad estimate becomes very smal in polynomial time. (This would stil hold if 3/4 were replaced by any fied constant strictly greater than 1/2.) More precisely, we have

of problems (see ¡98J for a surey), and there has also been considerable interest in randomised technques for combinatorial optimisation ¡60J.

Proposition 1.2 If there exists a polynomially time-bounded randomised

Following Stockmeyer ¡91J and Karp and Luby ¡56J, we use these ideas to establish a weaker notion of tractabilty for counting problems. Evidence

approximate counter C for R within ratio p, then there exists a PTM C'

that apparently hard counting problems may be tractable in this sense is provided by the latter authors, who exhbit a fuy-polynomial randomised

output and satisfies

approximate counter (see below) for satisfyng assignents of a DNF for-

mula (example 1 of the previous section).

which on inputs (x,8) E L* X iæ+ always halts with non-negative real-valued

Pr( C(x, 8) approximates #R(x) within ratio p(lxl)) 2: 1 - 8, and whose runtime is bounded by a polynomial in ixl and 19 8-1.

If a, â and r are non-negative real numbers with r 2: 1, we say that

â approximates a within ratio r if âr-1 :: a :: âr. Let 9 be a non-negative integer-vaued function on L* and P a real-vaued function on N such that p(n) 2: 1 for al n E N. A randomised approximation algorithm for 9 within ratio p is a PTM C which on inputs x E L* always halts with non-negative real-vaued output and satisfies Pr( C(x) approximates g(x) within ratio p(lxl)) 2: 3/4. If C is in fact deterministic, then it is an approximation algorithm for 9

within ratio p. In the case that 9 is the counting function #R for some

Proof: The required procedure C' makes p(lg 8-1) cal to C, with input x, for a suitable polynomial p and returns the median of the values obtained.

For the detais, see Lemma 6.1 of ¡53J. D Clearly, the powering operation of Proposition 1.2 may be applied in an identical maner to a f.p. randomised approximate counter, which wil constitute our primary notion of tractabilty for counting problems.

within ratio Pi if C is deterministic and p( n) = 1 for al n E N then C is

We now turn our attention to unform generation problems, takng our notions of tractabilty from Jerrum, Valant and Vaziran ¡53J. It is tempt-

an exact counter for R. In al cases, C is polynomially time-bounded if its

ing to demand that an exactly uniform generation algorithm should always

runtime is bounded by p (Ixl) for some polynomial p and al inputs x E L*.

halt and output some element of the solution set, assuming the latter is

relation Rover L, we cal C a (randomised) approximate counter for R

Ideally, we would like to be able to specify the factor in the approximation as par of the input, so that arbitrary accuracy can be achieved

(usualy at the cost ofincreased computation time). We say that a PTM C

is a randomised approximation scheme for 9 if on inputs (x, e) E L* x iæ+ it always halts with non-negative real-vaued output and satisfies

non-empty. However, this would not be reasonable within our model since any halting computation of a PTM is performed with probabilty 1/2t for some tEN: such a machine would thus not be capable of unformly generating (say) the elements of a solution set of cardinality 3 in finite time. We therefore allow generators to fail (i.e., produce no meanngf output) with bounded probabilty. Once the possibilty of failure is admitted, a number

Pr( C(x, e) approximates g(x) within ratio 1 + e) 2:3/4.

of attractive and natural generation paradigms become avaable, some of

If C is deterministic, it is an approximation scheme for g. If 9 = #R, we

As in the case of counting, we also introduce a relaxation of the original problem by alowing the output distribution to deviate from uniformity by

speak of a (randomised) approximate counter for R. C is fully-polynomial

which we shall describe in due course.

6

1.2 NOTIONS OF TRACTABILITY

1.2 NOTIONS OF TRACTABILITY

, specified amount, caled the tolerance. The strict uniformity requirement ) seldom of practical importance and if the deviation is smal enough it wil ie effectively undetectable. Furthermore, our notion of "alost uniform" :eneration arises naturally in a number of ways: as a problem which is iolynomial time equivalent to randomised approximate counting for selfeducible relations (Section 1.4); as a generalsation of uniform generation

"hich is robust with respect to extensions of the randomised model (Secion 1.3); and, most importantly, from dynamic stochastic algorithms for ;eneration problems (Chapter 2 onwards).

Let R be a relation over E, and € : N -+ IR+ U rO)-. A PTM Q is an ~lmost uniform generator for R within tolerance € iff its output on input : E E*, when defined, belongs to the set E* U r?)- and satisfies

(gl) If R(x) -I ø then Pr(Q(x) E E*) 2: 1/2.

I í

I I

17

where the input (x, €) E E* X IR+. Q is then fully-polynomial if its runtime is bounded by a polynomial in ixl and 19 c1. Note that the definition of f.p. here diers from that for approximate counters in the presence of the

logarithm: the errors tolerated in generation are exponentialy smal. (The

vaue C1 may be thought of as being specified on the input tape as an integer in binary notation.) As before, f.p. generators are assumed always to halt within their time bound.

Clearly, if R(x) -I ø the probabilty offaiure of any ofthe time-bounded generators defined above may be reduced rapidly by repeated trials. More precisely, Ig8-1 iterations of the generator sufce to ensure that the failure probabilty does not exceed 8. By the same token, we could replace 1/2

in the definition by l/p(lxl) for an arbitrary polynomial p. Futhermore, if the construction problem for a relation R is solvable deterministicaly in polynomial time then a f.p. almost uniform generator for R may be modifed so as never to fail when R(x) is non-empty.

(g2) There exists a function ø : E* -+ (0,1) such that, for al y E E*,

y tt R(x) =? Pr(Q(x) = y) = 0; Y E R(x) =? (1 + €(Ixl)) -1 Ø(x) :: Pr(Q(x) = y) :: (1 + €(Ixl) )Ø(x).

We shall regard the existence of a f.p. generator as an effective criterion

for tractabilty of generation problems. We should point out that the gap between exact and approximate notions here is apparently considerably smaller than in the case of counting. For example, viewed as black boxes,

.rote that the generator may only output valid solutions or the failure

alost uniform and uniform generators wil be effectively indistingushable

ymbol ?, and that it either fails or runs forever when the solution set s empty. Q is a uniform generator for R if €(n) = 0 for all n E N. In

under any reasonable definition of a statistical test involving polynomialy many observations. Moreover, we know of no structures which can be

ither case, Q is polynomially time-bounded if its runtime is bounded by a iolynomial in Ixl.1 If Q is polynomialy time-bounded, we may attach a

generated alost uniformly in an effcient manner but for which exactly

:lock to it and force al non-accepting computations to halt within the time

in Section 1.5, such a dichotomy does arise in the case of approximate

iound with output ? We wil assume that all such generators have been

counting.

uniform generation is (in some appropriate sense) hard. As we shal see

Ldapted so as to have this property.

By analogy with counters, it should be clear how to define an almost iniform generator Q for R, whose tolerance € :; 0 can be preset as part of

he input. Specifcaly, (g2) above now becomes (g2') There exists a function ø : E* x IR+ -+ (0,1) such that, for al y E E*,

y tt R(x) =? Pr(Q(x,€) = y) = OJ Y E R(x) =? (1 + €)-1ø(X, €) :: Pr(Q(x, €) = y) :: (1 + €)Ø(x, E),

For future reference, we also note here a standard randomised definition of

tractabilty for existence problems. The existence problem for a relation R ç E* x E* belongs to the class RP ("randomised polynomial time") . if there exists a PTM M with polynomialy bounded runtime which behaves as follows: for all inputs x E E*, R(x) -I ø =? Pr(M accepts x) 2: 1/2; R(x) = ø =? Pr(M accepts x) = O.

The probabilty of M makng an error is therefore one-sided, and can be i In many contexts, an appropriate complexity measure for a PTM is its exiected runtime. In the case of generation, however, a bound on the expected untime leaves open the undesirable possibilty that certain solutions only ever Lppear after a very long time. For approximate counters, either measure wil do.

made to decay very rapidly by the standard device of repeated trial. The

class RP (alo caled VPP by Gil (39)) replaces P as the class of tractable decision problems for randomised computation. Note that P ç RP ç NP, and, since problems in RP are intuitively tractable, the second inclusion is

1.3 AN EXTENDED MODEL

g

1.3 AN EXTENDED MODEL

ridely conjectured to be strict. The first is also thought to be strict for

i

ifferent reasons.

2. (Probabilstic execution)

with prob q do P

l

..3 An extended model I

3. (m-way branching)

rsing j successive tosses of its fair coin, a PTM can clearly simulate

either (with prob qo) Po

ist those. branching probabilties which are of the form i2-i, for i E

or (with prob qi) Pi

0,1,...,2i)-. The inabilty of a PTM to branch with more general probbilties often makes the design and justification of algorithms needlessly implicated. Indeed, randomised algorithms in the literature are typicaly icpressed in terms of more general branching probabilties involving the ttio of two previously

computed integers. Accordingly, in this section we

itroduce an extension of the PTM model which can easily simulate this lnd of branching behaviour. We go on to establish sufcient conditions

ir counting and generation algorithms in this model to be effciently imlementable by a PTM. This wil allow us in the sequel to work with the dended model while retaining our basic definitions of tractabilty. The roof details in this section are not central to the main development and iay safely be skipped.

An oracle coin machine (OCM) is a generalsation of a PTM in which

or (with prob qm-i) Pm-i

where m :; 1 and Li qi = 1. 4. (Weighted selection from an m-set)

select j E to, 1,..., m - 1)- with prob %1 Li qi where m :; O. (If Li qi = 0, the machie should halt with output?) . 5. (Uniform selection form an ordered n-set)

ie unbiased coin is replaced by a coin whose bias can be set and varied by

ie machine itself. The bias of the coin is determined by the contents of a iecial bias tape: if in a coin-tossing state this tape contains the encoding , a rational number q = rls, where r, s are coprime natural numbers . binary with r S sand s :; 0, then the two possible transitions are iosen with probabilties q and 1 - q respectively; otherwise, the machine

ilts immediately with distingushed output ? In all other respects, an

19

select j E to, 1, . . . , n - 1)- u.a.r. where n:; O.

Note that the integer elements in 4 and 5 may be viewed as indices, thus allowing selection from general ordered sets.

CM is identical to a PTM. The reason for rejecting such a machine as the :imary randomised model is that an elementary branching step could not

Proposition 1.3 Each of the above branching operations can be performed

asonably be implemented by a physical device in constant time.

by an OeM in runtime bounded by a polynomial in m and the sizes of the binary representations of q, tqi)- and n, plus the maxmum runtime of the programs P, tPi)-.

As is customary, Tuing machine algorithms wil be expressed in a diect of Pidgin AlgoL The OCM model alows this language to be augented with the probabilstic branching commands defined below, whose mantics should be obvious. In what follows, q, tqiÌ denote rationals in

Proof:

ie interval ¡0,1J, n, m natural numbers, and P, tPi)- programs.

1. This is the basic OCM branchig operation. The time requied, in 1. (Two-way branching)

addition to that for Po and Pi, is a single branching step plus the

time to write the encoding of q on the bias tape. either (with prob q) Po

or (with prob 1 - q) Pi

2. A special case of 1 in which Pi is nul.

1.3 AN EXTENDED MODEL

20

1.3 AN EXTENDED MODEL

21

3. By performing appropriate tests and relabellng if necessary, we may

introduced earlier. (Analogous questions for existence problems with re-

assume w.l.o.g. that al the qi are greater than O. The OCM fist

spect to a similar extended model have been studied by Lautemann (61j.)

computes the quantities q~ = qd(1 - E~:~ qj) for a -: i -: m - 1,

In the following, we refer to OCMs which serve as counters or generators for

and qb = qo. It then executes the following sequence of two-way

a relation Rover E in the vaious senses aleady introduced: the definitions difer from those of the previous section only in the model of computation.

branchings:

First let us observe that, as far as our most important approximate either (with prob qb) Po

or (with prob 1 - qb)

notions of counting and generation are concerned, the class of tractable problems is robust with respect to the change of modeL.

either (with prob qD Pi

or (with prob l-qD

Proposition 1.4 If there exists a f.p. OCM randomised almost uniform

generator (respectively, a f.p. OCM randomised approximate counter) either (with prob q;"-2) Pm-2

or (with prob 1 - q;"-2) Pm-i Clearly, both the arithmetic and the branching can be performed in time bounded by a polynomial in m and the sizes of the representations of ~ qd. 4. This is essentialy equivalent to 3.

5. Consider the following recursive procedure which uses only two-way

branchings:

for R, then there exists a J.P. almost uniform generator (respectively, a

f.p. randomised approximate counter) for R.

Proof: Let 9 be the postulated OCM generator for R, and let p be a polynomial bounding its runtime. We wil describe a PTM g' which on input (x, €) E E* x :i+ approximately simulates the computation process of 9 on input (x, €j3).

Assume w.l.o.g. that € :: 1, and set m = max~p(lxl,lg(3j€)),41: then m bounds the runtime of 9 on input (x, €j3), and al non-trivial branching probabilties of 9 must lie in the range (2-m, 1- 2-mj. The idea is that, for

procedure select(a, b);

any such branching probabilty q, g' wi compute a good approximation

begin L := b - a + 1;

to q of the form iq2-j, where iq EN and j = 2m + Ig(3j€) + 1. Specifcaly, if we set iq = Lq 2j J, then the rational iq2-j and 1 - iq2-j approximate q

if L = 1 then j := a

and 1 - q respectively within ratio 1 + 2-m€j3, as may easily be verified.

else begin

The simulation proceeds as follows. Deterministic steps of 9 are simulated directly by g', and if 9 halts then g' halts, necessarily with the same output.

m := (a + b) div 2; l':= m - a + 1;

either (with prob l jl) select(a, m) or (with prob 1 -l' jl) select(m + 1, b)

end end The call select(O, n - 1) achieves the desired effect and leads to at

most O(log n) levels of recursion, while the arithmetic is triviaL. The total runtime is therefore polynomial in the size of the representation

ofn. D he next issue we have to address is the effective implementation of OCM .gorithms in the more restricted PTM modeL. We shal do this with ref'ence to the notions of tractabilty for counting and generation problems

To simulate a non-trivial branching step of 9 of the form either (with prob q) Po

or (with prob 1- q) Pi

(1.1)

9 tosses its fai coin j times and performs the branching either (with prob iqj2j) simulate Po

or (with prob 1 - iqj2j) simulate Pi The time requied to compute iq and to do the coin tossing is clearly bounded by a polynomial in m and 19 C i . Since there are at most m steps of 9 to simulate, the runtime of g' is bounded by a polynomial in ixl and 19 c i, as requied.

1.3 AN EXTENDED MODEL

1.3 AN EXTENDED MODEL

)w suppose that 9 performs some computation c with probabilty p(c). r our earlier remarks, the probabilty that gr simulates c, yielding the me output, approximates p(c) within ratio (1 + 2-mE/3)m :: 1 + m22-mE/3 :: 1 + E/3, iere we have used the binomial theorem and the bounds m ;: 4 and E :: 1. lus the tolerance in the output distribution of g' is at most (1 + E/3)2 ::

I- E. Applying the same argument to non-accepting computations, we see

at the failure probabilty of gr exceeds that of 9 by at most a factor of I- E/3 :: 4/3, and so is bounded above by 2/3. Thus g', modified to allow lingle repeated trial on failure, is a f.p. almost uniform generator for R. ie proof for the counter is identical, except that the final modification

ikes use of the powering operation of Proposition 1.2. D In very similar fashion, we can show the following.

23

(Note that the approximate simulation in the proof of Proposition 1.4 could

also be formalsed in this way, though 'I(x) would have to be replaced by

i l

a smal range of vaues.) Of course, this is a minimal requirement since it

demands only that the output distribution of M (conditional on acceptance) be preserved. The simulation is effcient in the sense that the runtime of Mr the failure probabilty of M is not much greater than that of M, and that if is bounded away from 1 then that of M' may be similarly bounded using only a polynomial number of repeated trials. The following observation is immediate from the above definition.

Proposition 1.6 Suppose that M is a polynomially time-bounded OCM

uniform generator for R, and that the PTM Mr effciently simulates M. Then M', suitably powered to reduce its failure probability, is a polynomially

time-bounded uniform generator for R. D Next we derive a simple sufcient condition for effcient simulation by a PTM to be possible. The condition is expressed in terms of the probabilty distribution over the accepting computations of the OCM.

'oposition 1.5 If there exists a polynomially time-bounded OCM almost iform generator for R within tolerance E( n) (respectively, a polynomiy time-bounded OCM randomised approximate counter for R within ra-

Theorem 1. 7 Let M be an OCM with polynomially bounded runtime. Suppose there exists a polynomial time computable function N : E* - N+

i p( n)), then there exists a polynomially time-bounded almost uniform rierator for R within tolerance E(n) + 2-p(n) for any polynomial p (respec'€y, a polynomially time-bounded randomised approximate counter for R

N(x) Pr(CompM(x) = c)

thin ratio p(n)). D :act generators differ from the above devices in that their output distribun is prescribed precisely up to the probabilty of failure, so the straightward approach of the previous proof is not enough here. Before dealing Gh this problem, it wil be usefu to formalse the concept of simulation the present context.

Let M be an OCM and T(x) its runtime on input x. A PTM Mr is said effciently simulate M iff

:sl) For al inputs x for M and al strings y -l ?, Pr(M'(x) = y) = 'I(x) Pr(M(x) = y) , where 'I(x) ;: l/pi(lxl) for some polynomial Pi. 82) For al inputs x for M, the runtime of Mr on x is at most P2(lxl)T(x)

for some polynomial P2.

with the property that, for all inputs x E E* and all accepting computations c of M on input x, the quantity

is integral. Then M can be effciently simulated by a PTM. Proof: Let p be a polynomial bounding the runtime of M: since we are

only required to simulate accepting computations, we may assume that all non-accepting computations of M halt within this time bound with output ? Let x be an input for M, and set m = p(lxl). Clearly, al nontrivial branching probabilties of M on this input must lie in the range (2-m, 1- 2-m). The required PTM M' operates in two consecutive phases: a simulation phase and a correction phase. The first of these is very similar to the simulation in the proof of Proposition 1.4, except that Mr must be careful to select its branching probabilties from a smal set so that

subsequent correction is possible. Specifically, M' works with the set

A = U2i/2m12-2m : 1 :: i :: 4m2l of cardinality 4m2. It is not hard to see (53, Lemma 3.2) that for each real q E (2-m, 1 - 2-m) there exists aq E A such that

aq :: q :: (1 + l/m)aq,

(1.2)

--. ---~---'-----

I;

24

1.3 AN EXTENDED MODEL

I:

so that in Paricular aq approximates q within ratio 1 + 11m.

25

where 'Ø( x) is independent of e. For the simulation to be effcient, we require

As before, M' simulates deterministic steps of M directly. However, each non-trivial branching of the form (1.1) is now simulated by M' using the branching

a lower bound on 'Ø, which we get by choosing k appropriately. Recal that

the only constraint on the choice of k is that q :$ 1. But from (1.3) we have f

either (with prob aq) simulate Po

r

q = ;~~ 'Ø(x) :$ e'Ø(x),

i'

or (with prob ai-q) simulate Pi

or (with prob 1 - aq - ai_q) halt with output? II

From the definition of A, this can be realsed by M' using only 2m tosses of its fair coin. Assuming no faiure, the simulation phase continues until A1

1.3 AN EXTENDED MODEL

!

I

so we may select k to be maxmal such that 'Ø(x) = ?r(A)N(x)2k :$ lIe. It then follows that 'Ø(x) ). 1/2e is bounded below as required.

to be described shortly. The total time requied for the simulation phase

The above discussion indicates that M' satisfies condition (sl) of the definition. For condition (s2) we need only make the additional observation that the arithmetic of the correction phase can be performed in polynomial time since the cardinalty of A is only 4m2. Hence the runtime of M' is

is clearly bounded by a polynomial in Ixl.

polynomialy bounded as required. 0

reaches a halting confguation. If the corresponding output of M is ? then M' also halts with output ?; otherwise, M' enters its correction phase,

I: i I: l i

Upon entry to the correction phase, M' has simulated some accepting com-

putation e of M. By maintaining two running products, we may assume that M' has computed both the probabilty pee) = Pr(CompM(x) = e) (i.e., the product of the branching probabilties it has simulated), and the probabilty p'(e) = Pr(CompMI(x) simulates e) (i.e., the product of the branching probabilties actualy used). By (1.2), each term in the second product approximates the corresponding term in the fist within ratio 1 + 11m, and we have p'(e) :$ pee) :$ (1 + 1/m)m p'(e) :$ ep'(e). (1.3) Futhermore, M' is able to compute in polynomial time a natural number N(x) such that N(x)p(e) is integral, as stipulated in the statement of the

theorem.

The sufcient condition of Theorem 1. 7 may at first sight appear cumbersome to apply in practice. However, most known generation algorithms

can readily be seen to satisfy it. One typical case is an OCM uniform generator each of whose accepting computations yields a different solution as output. For a fied input x, any such computation must therefore be

executed with probabilty rxlsx for constants rx, Sx which are known a posteriori, and we may take N(x) = sx' A rather less trivial example is provided by the following algorithm, due to Nijenhuis and Wilf (78), for uniformly generating partitions of a positive integer.

Example 1.8 A partition of a positive integer n is a representation ?r of the form

Denote by ?reA) the product IIaEA am. The correction phase of M' now consists of a single branching of the form q ;= ?r(A)N(x)2kp(e)lp'(e);

either (with prob q) halt with output y

or (with prob 1 - q) halt with output? where y is the output of the simulated computation e and k is an integer independent of e to be specified below: in particular, k wil be chosen so that q :$ 1. It should be clear from the form of q that the above branching can be achieved by M' using its fair coin. Fuhermore, the probabilty that M' simulates e and, after correction, accepts with the same output is (?r(A)N(x)2k) pee) = 'Ø(x)p(e),

n = ri + . . . + r¡ for integers ri with ri ~ r2 ~ . . . ~ r¡ ). O. We shal identify partitions of n with multisets of the form lJLi . ri, . . . , JLk . rd, where JLi, ri E N+, the ri are distinct and ¿~i JLiri = n. For convenience, n is alowed to be zero,

in which case the only partition is the empty set. Define the relation II which associates with each n E N the set of partitions

of n, and consider the recursive procedure Genpart of Figure 1.1. Here ?r + j . r denotes the operation of adjoining j copies of r to the multiset ?r. It is assumed in line (2) that the counting function #II for II is computed by some other procedure (using, e.g., the recurrence relation implicit in Genpart itself), and that #II(m) = 0 for m -( O.

We claim that the cal Genpart(n, ø) uniformly generates partitions of n. More generalv. Wp "li"m h.. :_ _L _, .

26

1.3 AN EXTENDED MODEL

1.3 AN EXTENDED MODEL

27

that c is executed with probabilty ¡ procedure Genpart(n : naturaL

number; 11 : partition);

begin

p(c) =mit¡#II(mi) #II(m2) #II(m3) t¡ #II(O) = ~i=iIImi ~ m2t2#II(m2) m¡...#II(m¡) #II(n)

(i) if n = 0 then halt with output 11

else begin

for some ti,mi,l E ii+, with n = mi :; m2 :; ... :; m¡ :; O. Hence

(2) for (r,j) E ¡i,n) x ¡i,nj do q(r,j) :=r#IT(n-jr);

the product n! #II( n) p( c) is always integraL We may therefore take

(3) select (r,j) E ¡i,n) x ¡i,n) with prob q(r,j)lE,q(r',j');

N(n) = n! #II(n) in Theorem 1.7 and deduce that the algorithm has a

(4) Genpart(n - jr, 11 + j . r)

PTM implementation. D

end end

Figue 1.1: Algorithm for uniformly generating partitions

To conclude this discussion of models, we futher motivate Theorem 1.7 by showing that effcient simulation by a PTM is not possible for arbitrary polynomialy time-bounded OCMs. In other words, the extended model is strictly more powerfu than the PTM model in terms of its abilty to realse a given output distribution exactly in polynomial time.

uniformly generates partitions of n and adjoins them to 7r. The base case n = 0 is immediate from line (1). Suppose then that n :; 0 and consider

Theorem 1.9 There exists a polynomially time-bounded OeM which can-

the partition 7r' of n given by

not be effciently simulated by any PTM.

7r' = tILi.ri,...,ILk 'rkì.

Examination of the algorithm reveals that 7r' may be adjoined to 7r in precisely ILi + . . . + ILk ways as follows: in line (3) select r = ri for some

1 :: i :: k, and j in the range 1 :: j :: ILi; finally, adjoin to 7r + j . ri the partition

Proof: Consider the OCM M which, on any input x of length n, executes the following: select j E tI,..., 2nì u.a.r.j

either (with prob 1fj) halt with output? or (with prob 1 - 1fj) halt with output j

tILi' ri,..., (ILi - j). ri,... ,ILk' rkÌ

of n - jr i via the recursive call in line (4). By the inductive hypothesis, the recursive call generates partitions unformly. In view of lines (2) and (3), the probabilty that 7r' is appended to 7r is therefore

'"k l-¡ '" ri#II(n ')- 1Jrik

'" n L. ~ Q(n) #II(n - jri) = Q(n) ~ ILiri = Q(n) ,

o=i 3=i o=i

By Proposition 1.3, the runtime of M is bounded by a polynomial in n. Fu-

thermore, for each j in the range 1 :: j :: 2n, M outputs j with probabilty (j - 1)(2nj)-i. Note incidentaly that M is effective in the sense that its faiure probabilty on inputs of length n is 1 - H2n /2n, which tends to zero

exponentialy fast with n. (Here Hk denotes the kth harmonic number.)

where Q(n) = ¿q(r',j') as in line (3) and depends only on n. Since 7r'

Now suppose there exists a PTM M' which effciently simulates M. Since M is polynomialy time-bounded, M' must be also, by condition (s2). So let p be a polynomial bounding its runtime, and set m = p(n). Then clearly, for

was chosen arbitrarily, we conclude that the generation is uniform and also

any input x of length n and integer j, we must have

that Q(n) = n#II(n). For a combinatorial derivation of this algorithm,

Pr(M'(x) = j) = i(j)Tm

the reader is referred to ¡78J.

It should be clear that Genpart can be implemented by a polyno~ially time-bounded OCM. (Note that the evaluation of #II at the appropriate points can be performed effciently.) Now let c be any computation of such a machine on input (n, ø) with n :; O. The form of the selection probabilties in line (3), together with the fact that Q(m) = m#II(m) for all m, implies

for some integer i(j) in the range ¡O,2mJ.

Given the output distribution of M, condition (s1) of the definition therefore demands that i(j) = r¡(x) j -- 1

J

(1.4)

28 1.4 COUNTING, GENERATION AND SELF-REDUCIBILITY

I

for 1 ~ j ~ 2n and some constant r¡(x) :; 0 independent of j. Setting j = 2 in (1.4) implies that r¡(x) = 2i(2), so r¡(x) is integral and bounded above by 2m+!. FUhermore, since i(j) itself is integral, (1.4) also implies that j I r¡(x) for 1 ~ j ~ 2n. However, it is well known that, for infnitely many natural numbers r, lcm-(I,... ,r) 2: Aer for some constant A (see, e.g., ¡32, page 34)). We must therefore have

i

r¡(x) 2: lcm-(I,..., 2n) 2: ke2n

infinitely often, which is a contradiction since r¡(x) is bounded above by 2m+! = 2P(n)+1, and p is a polynomial. We must therefore conclude that

1.4 COUNTING, GENERATION AND SELF-REDUCIBILITY 29 procedure GenR(v : vertex);

I

begin

i!

i

(1) z := inst(v); (2) if lR(z) = 0 then halt with output sol(v)

else begin

i

(3) select w E EO"(z) with prob C('l(z,w))/Lw'C('l(z,w')); (4) inst(u) := 'l(z,w); sol(u):= sol(v). w;

(5) GenR(u) end end

the integers i(j) do not always exist as claimed, so M' does not effciently

simulate M. D Figue 1.2: Reduction from generation to counting

It could be argued that the above counterexample exploits the special properties of a bizarre output distribution, and that a similar result would not necessarily hold if attention were restricted to (say) uniform generators.

However, we claim that our weak definition of simulation makes it entirely reasonable to consider general distributions. This is because we are really only using outputs to encode distinct computations of the OCM. Accord-

ing to our definition, the algorithm of the simulating machine need bear no resemblance to that of the original OCM. A more realistic definition of simulation would requie that each individual computation step be simulated

explicitly. (Note that the simulations of Proposition 1.4 and Theorem 1.7 have this property.) Under this definition, the proof of Theorem 1.9 stil holds even when the OCM M is modifed so as to have a trivial output distribution. The question of whether there exists a relation which can be uniformly

generated in polynomial time by an OCM but by no PTM is open, and seems dicult. In view of Theorems 1.7 and 1.9, this is not inconceivable but any effcient OCM generator for such a relation must necessarily induce

a fairly complex distribution on its set of accepting computations.

1.4 Counting, generation and

self- red ucibility In this section we describe two reductions due to Jerrum, Valant and Vazirani which together establish a close relationship between the complexities of approximate counting and almost unform generation for most interesting p-relations. In each case, the additional property which is needed to

make the reduction work is self-reducibilty. Similar results in a less general setting have been proved by Broder ¡16).

Most known unform generation algorithms for combinatorial structures may be viewed as instances of the followig generic reduction to the corresponding counting problem. Suppose that the structures have a simple

inductive construction, or more formaly that they may be described in terms of a self-reducible relation R, and let x be a problem instance with non-empty solution set. Now select a random path from the root of the

tree of derivations TR(x) to a leaf (solution), choosing the next edge at each stage with probabilty proportional to the number of solutions in the maxal subtree rooted at its lower end: thi inormation may be obtained by evauating the fuction #R for appropriate problem instance labels in the tree. It is then easy to see that the distribution over solutions is uni-

form. A formal specifcation of the algorithm is provided by the recursive

procedure GenR shown in Figue 1.2, in which 0', 'l, sol and inst have the meanings ascribed to them in the earlier definitions and C is an exact counter for R, viewed as an oracle. Elements of R(x) are generated using the cal GenR(u), where u represents the root of TR(x), i.e., inst(u) = x and sol( u) = A. Since the depth and vertex degree of the tree are polynomialy bounded, the algorithm can be implemented on a polynomialy time-bounded OCM equipped with the oracle C for #R. Futhermore, as

each computation yields a dierent output, Theorem 1.7 ensures that a PTM implementation exists. The above procedure can actualy stil be made to work when the count-

1.4 COUNTING, GENERATION AND SELF-REDUCIBILITY

30

r

ing information supplied by the oracle C is slightly inaccurate, specifically

i

if it is within ratio 1 + O(n-kR), where kR :; 0 is a constant satisfying

!

lR(X) = O(lxlkR) (recall condition (srI) in the definition of self-reducibilty).

~

This is done by appending a correction step prior to the output in line (2),

~

1.4 COUNTING, GENERATION AND SELF-REDUCIBILITY 31

of this event on the output probabilty p(y) of any solution y is thus an additive term :l8q(lxl), i.e.,

fj(x) - 8q(lxl) ~ p(y) ~ fj(x) + 8q(lxl).

much as in the proof of Theorem 1.7. To see how this works, view the uncor-

rected procedure as an approximate simulation of the ideal procedure with exact oracle: given the constraint on the counting estimates, each branching probabilty in line (3) is simulated within ratio (1 + cjlR(1/(X,w))2 for some fied constant c E N+. Taking the product of such factors along any path from root to leaf, and bearing in mind that the length function lR de-

~

(1.8)

(Note also that, since the oracle no longer decides with certainty whether a given solution set is non-empty, the process is not now confined to the tree of derivations but may occasionaly fal into other parts of the selfreducibilty tree. As a result, some non-solution leaves may be reached

with non-zero probabilty: however, since R is a p-relation the condition

creases strictly along the path, we find that the generation probabilty p(y)

(x, y) E R may always be checked prior to output.) Now observe that, since

of each solution y E R(x) approximates the ideal vaue Ij#R(x) within

solutions are strings oflength mover E, their total number #R(x) cannot exceed IElm. Combining this with the bound (1.7) on fj(x), we see that by

ratio

IR(x)-1 ( ) 2c

g (1 + ~r ~ n (1 + ~) = lR(X)2C .

choosing (1.5)

(1 + cjm)2IElmq(lxl) - q(lxl)

Writing m = lR(X), this implies that p(y) is bounded below by

m-2c m-2c

p(y):; -:; =fj(x). - #R(x) - i. , ~Lu\r"/u\'

8 = (€j2)m-2c .( (€j2)fj(x) ,

(1.6)

The correction step is now simply a matter of outputting y with probabilty

fj(x)jp(y), which by (1.6) is ~ 1, and failng otherwise. Each solution is thus generated with the unform probabilty fj(x), and the probabilty that no faiure occurs is m-2c #R(x) fj(x) ~ (1 + cjm)2 ' (1.7) which may be boosted to Ij2 as usual using only polynomialy many re-

peated trial. Implementation in the PTM model follows as before. Finaly, consider what happens if we use the same procedure when C is a randomised approximate counter within the above ratio. Assuming initialy that all vaues returned by C are within this ratio, the previous arguent stil holds and solutions are generated with some unform probabilty fj( x) which is again bounded as in (1.7). Unfortunately, we can say nothig about the output distribution when some vaue returned by C happens to be very inaccurate: however, by appealg to Proposition 1.2, we may ensure that C

we can write the additive approximation (1.8) of fj(x) by p(y) as a relative approximation within ratio 1 + € (assuming 0 .( € ~ 1), so that the resulting generator is almost uniform within tolerance €. Since the powering operation requies time polynomial in 19 8-1, the runtime of the procedure with C as oracle is polynomial in Igc1 and lxi, as required. Implementation on a PTM follows directly from Proposition 1.4. If C itself is polynomially time-bounded, then the entire procedure constitutes a f.p. almost uniform

generator for R. The above discussion is summarised in the following theorem.

Theorem 1.10 (Jerrum, Valiant and Vaziran) Let R be a self-reducible relation over E and kR a constant such that, for all pairs (x, y) E R, lyl = O(lxlkR). If there exists a polynomially time-bounded randomised

approximate counter for R within ratio 1 + O(n-kR) then there exists a f.p. almost uniform generator for R. Moreover, if the counter is deterministic then there exists a polynomially time-bounded uniform generator

for R. 0

behaves badly with such smal probabilty 8 that the resulting procedure

Note that the ratio I+O(n-kR) appears to be a threshold value for this simple reduction technique: under a weaker hypothesis, there seems to be

(with C as oracle) is a f.p. alost unform generator for R.

no polynomial bound on the product (1.5), indicating that the cumulative

To see how to choose 8, note fist that the total number of oracle call

is bounded by q(lxl), where q is a polynomial which depends on the depth and vertex degree of the tree. The probabilty that any returned vaue fal outside the acceptable range is therefore at most 8q(lxl), and the effect

errors may then become too large for effective correction to be possible. As we have already mentioned, numerous generation algorithms may readily be derived from the above reduction to exact counting. Several

simple examples appear in the book by Nijenhuis and Wilf (78J, who also

32 1.4 COUNTING, GENERATION AND SELF-REDUCIBILITY

present an alternative formulation of the generic reduction in this case. (Note that the algorithm of Example 1.8 can be recast in this form by

modifying the underlying relation and biasing the generation probabilties appropriately.) Other examples in the literature include generating spanning trees in a graph (40), terminal strings of given length in an unambiguous context-free grammar (44), unabelled trees (99) and labelled connected graphs (79). Tyicaly, the counting information is derived from

a recurrence relation which alo defines a self-reducibilty for the structures in question: occasionaly, however, a deeper result is involved (e.g., in the case of spanning trees, Kirchoff's matrix tree theorem). We give one more example below, and thereafter in this monograph concentrate on cases where exact counting is apparently not possible.2 Generation algorithms which make use of approximate counting infor-

mation and the correction technque are much less common: a notable exception is the elegant method of Bach (8) for uniformly generating factored

integers in a given interv. The approach can also be used to refine the scheme proposed by Dixon and Wil (26) for generating unabelled graphs. (Their algorithm has polynomialy bounded expected runtime, but the distribution induced by any polynomial time truncation has very large bias. The refinement guarantees a unform distribution in polynomial time.) We wil not expand on this example here but instead refer the reader to a more elegant attack on the same problem by Wormald (101). The following little example is chosen partly because it ilustrates the

reduction to exact counting and parly because of its superficial resemblance to the problem of generating labelled graphs with given vertex degrees,

which we shal discuss at length in Chapter 4.

Example 1.11 Consider the relation TREES which associates with each sequence g = (gi)iEI of non-negative integers the set of labelled trees with vertex set I in which vertex i has degree gi. Clearly, if TREES(g) is non-

empty then gi )- 0 for al iE I and EiEI gi = 21I1 - 2: we call g valid in this case. If g is vad then gio = 1 for at least one io E I, which suggests a natural self-reducibilty for the relation TREES. By considering al possible

I i

1.4 COUNTING, GENERATION AND SELF-REDUCIBILITY 33 neighbours of io, we may write

~ '"

J

I~

.. ,-,

TREES(g) = U t T U Hio,j)l : T E TREES(gU)) J ' (1.9) jEI\fioÌ

..

i

where gU) is the degree sequence with vertex set I \ r io 1 satisfying

g.' = i ifi-J'. , -, (') tg.-1

i gi, otherwise,

and we have identifed trees with their edge sets. To generate trees on g uniformly, it is therefore sufcient by Theorem 1.10

to find some effcient means of counting them. This is provided by the following formula, in which g is assumed vad:

(III - 2)! #TREES(g) = IliEi(gi - I)! . The formula is easily verifed by induction on III using the recurrence im-

plicit in (1.9) (see, e.g., (64, Problem 4.1)). D The converse reduction from counting to generation, while perhaps less familar, is equaly intuitive. Clearly, a counting procedure based on random generators canot be deterministic, so we aim for an effcient ran-

domised approximate counter. Again we make the assumption that R is self-reducible and work with the tree of derivations for a problem instance x

with R( x) =l ø. The task at hand is therefore to estimate the number of leaves in a rooted tree T given an aÌost unform generator for the leaves in any maxmal subtree. For such a subtree 8, let L(8) denote the number of leaves in 8. The idea is to generate leaves of T and compute the proportion s of this sample which belongs to some subtree 8 rooted at a child of the root of T. Assuming that s is a reasonable estimate of the ratio L(8)/ L(T), we get an approximation for L(T) by recursively estimating L(8) and multiplying the result by S-1. The qualty of the estimate s at each stage depends on the size of the sample and the way in which the subtree 8 is selected. Clearly, higher

accuracy wil be achieved if the ratio L( 8) / L(T) is large, so we adopt the 2Uniform generation can sometimes be streamlined if the structures can be

policy of choosing 8 so as to maxmise the estimate s for the given sample.

"such a way that integers can be effciently mapped to structures. (This operation is

The following straightforward piece of statistics tell us how large a sample

placed in I-I-correspondence with a set of integers, typically il,..., NL, in

is requied in order to achieve a specifed accuracy.

called unranking.) Then it is enough to generate an integer u.a.r. and output the corresponding structure. As a non-trivial example, (21) develops this approach

Proposition 1.12 With the above notation, suppose that T has maximum

for spanning trees in a graph.

degree d and elements of the sample are generated almost uniformly within

34

1.4 COUNTING, GENERATION AND SELF-REDUCIBILITY

1.4 COUNTING, GENERATION AND SELF-REDUCIBILITY

tolerance e E (0,1). Then for any 8 E (0,1), the sample size t required to ensure that

i

Pr(s approximates L(S)IL(T) within ratio 1 + 5e) ~ 1- 8

i

is at most 9d3(e8)-1. 0

35

TI:= 1; while lR(x) ? 0 do begin make 3t calls of the form Q(x, €/lOm);

if at least t of these yield a solution then let Y = tY1, . . . , Yt 1 be the first t solutions

I

else halt with output 0;

A minor problem arises from the fact that the generator may fail to

for w E EtT(z) do s(w) := ItY E Y : w is a prefi of yll/lYl;

supply sufciently many meaningf outputs within a reasonable number of trials. However, if failure occurs in anyone trial with probabilty no more than 1/2, it is easy to see that

let w be such that s(w) = maxw, s(w');

TI:= TIls(w); x:= 'l(x,w) end; halt with output TI

Proposition 1.13 For any t, the probability that 3t trials fail to yield at

least t outputs distinct from? is at most 31t. 0

Figure 1.3: Procedure for estimating #R(x)

We therefore perform 3t trials at each level of recursion and examine the

outputs of the first t successfu ones: in the unkely event that insuffcient

Figue 1.3, in which it is assumed that Q is an almost unform generator

successfu outputs are generated, we abort the counting procedure with

for R.

default vaue O.

A partial converse to Theorem 1.10 may now be stated as follows.

Suppose now that we wish to approximate L(T) within ratio 1 + € using

the above method. Let the depth of T be m and assume for the moment that the procedure is not aborted because of generator failure. Setting

e = €/I0m and 8 = 118m in Proposition 1.12, we can fi t so that at each level of the tree the estimate s is accurate within ratio 1 + €/2m with probabilty at least 1 - 118m. Since there are at most m levels, with

Theorem 1.14 (Jerrum, Valiant ànd Vazirani) Let R be a self-reducible relation over E. If there exists a J.P. almost uniform generator for R then there exists a f.p. randomised approximate counter for R. 0 Combining Theorems 1.10 and 1.14, we arrive at

probabilty (1 - 1/8m)m ~ 7/8 the product of the factors s-1 therefore approximates L(T) within ratio (1 + €/2m)m :: 1 + € (assuming € :: 1).

Corollary 1.15 For a self-reducible relation Rover E, the following are

Finally, to dispense with generator failure, we assume futher that t ~ 24m

equivalent:

and deduce from Proposition 1.13 that the procedure runs to completion with probabilty at least (1 - 1/8m)m ~ 7/8. The procedure therefore

approximates L(T) within ratio 1+€ with probabilty at least (7/8)2 ~ 314, as required. Given an oracle which generates leaves alost uniformly within tolerance €/lOm, the runtime of the procedure is bounded by 3mt, which in turn by Proposition 1.12 is bounded by a polynomial in d, m and c1.

Returning to the case where T = TR(x) as above, note fist that the depth m and degree d of the tree are both polynomialy bounded in 1xJ The generation of leaves in any maxmal subtree may be accomplished'by an alost uniform generator for R within tolerance €/I0m. If the genera-

tor is f.p., its runtime on each trial wil then be bounded by a polynomial in ixl and ci, and thus the runtime of the entire procedure wil be sim-

ilarly bounded. The algorithm for estimating #R(x) is given in detail in

(i) There exists a J.P. almost uniform generator for R.

(ii) There exists a f.p. randomised approximate counter for R. (iii) There exists a polynomially time-bounded randomised approximate counter for R within ratio 1 + O(n-kR), where kR is a constant as above. 0 The implication (ii) =? (ii) in Corollary 1.15 indicates that the pair of reductions presented here actually yield a method for bootstrapping a polynomially time-bounded randomised approximate counter for R within the

threshold ratio 1 + O(n-kR) to one within ratio 1 + n-ß for any desired real ß. In view of the hypothesis of Theorem 1.10, however, bootstrapping of less accurate counters is apparently not possible using these techniques.

36

1.5 AN INTERESTING CLASS OF RELATIONS

Unfortunately, the reductions as they stand do not seem to yield a corresponding improvement mechanism for generators. This stems from the fact that the second reduction, une the first, demands of its oracle at

each level an accuracy which is not simply a function of the local problem instance, but which depends on the global quantities € and m: there is no analogue of the correction process employed in generation which would

allow larger errors to be handled. Consequently, we have to resort to a generator whose tolerance may be vaied.3 For future reference, however, we state a weaker version of Theorem 1.14 which presupposes only the

existence of a generator with fied tolerance. Theorem 1.16 For Rand kR as above, if

there exists a polynomially time-

bounded almost uniform generator for R within tolerance O(n-kR) then there exists a polynomially time-bounded randomised approximate counter for R within ratio O(nf;) for some constant ß.

Proof: Immediate from (1.5) and the proof of Theorem 1.14, setting e = c'lxl-kR for some constant c'. 0

1.5 AN INTERESTING CLASS OF RELATIONS

i i

37

In the remainder of this monograph, we wil make some contribution towards classifying natural relations according to the criterion of approximabilty, concentrating primarily on techniques for proving positive results. Before embarking on this investigation, however, we should be a

i

little clearer about the kind of relations for which this question is interesting.

Obviously, we are not particularly interested in relations, such as those mentioned in the previous section, whose counting problem is known to be

solvable èxactly in polynomial time. If the relation is self-reducible, this alo implies the existence of a polynomialy time-bounded uniform generator for it and the status of both problems is therefore fully resolved. Idealy,

we would lie some positive evidence of intractablility for the exact counting problem, which usualy means #P-completeness. (Note that there are

diffcult counting problems for which #P-completeness is not an appropriate notion of hardness. This is the case for most of the classical graphical

enumeration problems of the form: Given an integer n, compute the number of graphs of size n having a certain property l42). We refer to these as problems of unary type since they have only one input of each size. A more natural hardness concept here is that of #P i-completeness 197), which

We shal have a lot more to say about bootstrapping both counters and generators in Chapter 4, where we present a signcantly improved version of Theorem 1.10.

1.5 An interesting class of relations

unfortunately has proved rather hard to work with in practice.)

On the other hand, the question of approximablity is also trivial if the existence problem for the relation in question is suspected to be hard, in particular if it is NP-complete. To see this, observe that a polynomialy time-bounded alost unform generator for R within any tolerance € ~ 0 immediately yields an effcient solution to the corresponding existence prob-

In Section 1.2 it was argued that the counting problem for a p-relation R

lem in the RP sense defied earlier. If this latter problem were known to be NP-complete, then we could deduce that RP = NP, i.e., that the existence problem for any p-relation is tractable: this is widely held to be alost as

may be regarded as effectively tractable if there exists a f.p. randomised

unely as the assertion that P = NP. The existence of a polynomialy

approxiate counter for R. Simiarly, we may view the existence of a

time-bounded randomised approximate counter for such a relation within

f.p. alost unform generator for R as evidence for the tractablility of its unform generation problem. For the puroses of this discussion, let us cal a relation "approximable" if at least one of these problems for it is tractable. In most cases the relation wi be self-reducible, so in the light of Section 1.4 if one of the problems is tractable then so is the other.

any ratio p ~ 1 would alo imply that RP = NP. By Theorem 1.10, this is immediate if th~ relation is self-reducible. If not, we could stil use the counter to construct a polynomial time randomised algorithm with bounded two-sided error probabilty for an NP-complete existence problem associ-

ated with a self-reducible relation, such as SAT.4 The error may, however, be made one-sided in polynomial time by exploiting the self-reducibilty of

31n (53) it is informaly claimed that a generator satisfyng the hypothesis of

Theorem 1.16 can be bootstrapped to a f.p. almost uniform generator. This turns

4More precisely, we could locate the existence problem for SAT in the class

out to be true, as shown in Theorem 4.8, but the proof apparently does not follow

BPP (39). (SAT is the relation which associates Boolean formulae in conjunctive normal form with their satisfyng assigments.)

from the results of (53) and relies on the machinery developed in Chapter 4.

38

1.5 AN INTERESTING CLASS OF RELATIONS

1.5 AN INTERESTING CLASS OF RELATIONS

is trivial here since the empty set is always independent. We now proceed to show

SAT: before outputting a "yes" answer, it is first verified by explicit construction of a solution. This would imply that the existence problem for

SAT is in RP.

i

have aleady mentioned that many naturally occurring relations fall into this category: as well as perfect matchings in a graph and satisfying assign-

ments of a DNF formula, examples include matchings (independent sets of edges of any size) in a graph (97), directed trees and s-t-paths in a directed graph (97), and connected spanning subgraphs of a graph (81). For al these relations, the existence problem lies in P and the counting problem

is #P-complete.

Remark: We should mention by means of an example a further case in which uniform generation may become trivial. Consider the problem of uniformly generating labelled connected graphs with a given number of vertices. The following naïve procedure provides an effcient solution:

simply select a random graph of the required size and output it if it is connected, failing otherwise. The method works since almost every graph of each size is connected (see, e.g., (42)). (A non-trivial "exact" solution to this problem, based on the reduction to counting, also exists and can be found in (79).) 0

Having identified a potentially interesting class of relations, we should check that the question of approximabilty is a genuine issue for the class: in other words, we should be able to exhbit a natural relation in it which is approximable and another for which even approximate counting and gen-

eration are hard. A candidate for the fist of these criteria has aleady been mentioned, namely the DNF satisfiabilty relation. That this is approximable is shown by Karp and Luby (56), who give a f.p. randomised approximate counter for it. In fact, Jerrum, Valant and Vazirani (53)

prove the slightly stronger result that the relation has a polynomialy timebounded uniform generator. We do not describe these algorithms in detail here; the reader wil find an alternative approach to this problem in Example 4.4 of Chapter 4. Conversely, it is possible to find natural relations with easy existence and

construction problems which are nevertheless hard to approximate. This complexity gap presumably arises from an implicit requirement in counting and generation that al solutions be "equaly accessible". As an example,

consider the relation INDSETS which associates with an undirected graph G

al independent sets of vertices of G. Note that the construction problem

Theorem 1.17 If there exists a polynomially time-bounded uniform generator for INDSETS then RP = NP.

In view of these points, we shall focus on p-relations whose counting problem appears to be hard and whose existence problem is tractable. We

39

i Proof: Let is be the relation which associates with each graph G = (V, E)

and positive integer k al independent sets of G of size at least k. The existence problem for is is a standard NP-complete problem (38): we present a polynomial time reduction from it to the unform generation problem for INDSETS.

The reduction proceeds by replacing each vertex v of G by a cluster Cv of r independent vertices, where r wil be specified below. For each edge (u, v) of G, we add the set of edges H u', v') : u' E Cu, v' E Cv l, and denote the resulting graph G'.

Now let S be an independent set in G. An independent set S' in G' is a witness for S iff it satisfies tV E V : Cv n S' f: øl = S.

Note that every independent set in G' is a witness for a unique set in G. The proof hinges on the fact that, if r is chosen suitably, large independent sets have many more witnesses that smal ones. To see this, parition

the independent sets of G' into two classes, Large(G, k) and Smal(G, k), according to whether they are witnesses for sets in G of size ~ k or -( k respectively. Clearly, ILarge(G, k)1 ;: 0 iff G contains an independent set of

size k. So assume now that this holds, and for 0 ~ L ~ k let N¡ denote the number of witnesses for each independent set of size L in G. Then we have, from the construction of G',

N¡H ~ 2T N¡ for 0 ~ L ~ k . Let n be the number of vertices in G. Then G can contain at most (7) independent sets of size l, so k-l ( )

ISmall(G,k)1 ~ ~ 7 N¡ ~ 2nNk_l, since N¡ increases with l. Combining the above two inequalties gives ILarge(G, k)1 ~ Nk ~ 2T Nk-l ~ 2T-nISmall(G, k)l.

Hence by choosing r = n we may ensure that at least hal of the independent

sets in G' are witnesses for independent sets in G of size k or more. It follows that, if independent sets in G' can be generated uniformly in polynomial time, the existence problem for is lies in RP, as required. 0

40

1.5 AN INTERESTING CLASS OF RELATIONS

1.5 AN INTERESTING CLASS OF RELATIONS

Closer examination of the above proof reveals that a polynomialy time-

and existence; it turns out that this can be formalsed as follows. Recal that NP is the class of existence problems for p-relations and lies within the first

bounded almost uniform generator for INDSETS with very large tolerance (any polynomial fuction of the input size) would also be sufcient. Fu-

thermore, approximate counting even within a large ratio is easily seen to

i

level of the polynomial time hierarchy (90J. Fuhermore, the corresponding

I

following result, which is an application of work by Sipser (89J on unversal

be hard for the same reason.

The above technque of boosting the proportion of solutions which are

hard to detect was initialy used by Jerrum, Valant and Vaziran (53J to prove analogous hardness results for the relation of cycles in a directed graph; these transfer immediately to undirected graphs and s-t paths (i.e.,

simple paths between a specified pair of vertices). By parsimonious (or

41

class #P of exact counting problems is not known to lie within any fied level of the hierarchy (generalsed to Turing machines with output). The hash functions, indicates that the counting problem for any p-relation may

be solved approximately by a machine in the second level of the hierarchy, generalsed to include randomisation.

also seen to be hard to approximate: all vertex covers and al cliques in

Theorem 1.18 (Stockmeyer (91J) Let R ç E* x E* be a p-relation. Then there exists a PTM equipped with an NP oracle which is a J.P. randomised

a graph, satisfying assignments of a monotone Boolean formula in 2-CNF, and 3-colourings of dense5 graphs. The proof of Theorem 1.17, with minor modifcations, shows that the same holds for maxmal (i.e., non-extendable)

for R. D

alost parsimonious) reductions from INDSETS, the following relations are

independent sets, and thus also for minimal vertex covers and maxmal cliques (97J.

approximate counter for R. Furthermore, there exists a deterministic Turing machine equipped with a E~ oracle which is a J.P. approximate countei.

As noted in (53J, this immediately yields a similar characterisation for generation problems.

Remark: The proof of Theorem 1.17 ilustrates a feature of approximate counting of which the reader should be aware. We may view the

Corollary 1.19 For any p-relation R ç E* x E*, there exists a PTM with

above reduction as a proof that the exact counting problem for INDSETS

an NP oracle which is a J.P. almost uniform generator for R, and a PTM

is #P-hard: for with suitable choice of r, by evauating #INDSETS for the

with a E~ oracle which is a polynomially time-bounded uniform generator

transformed graph G' and looking only at the most significant figues in the output, #IS(G, k) can be computed exactly. The latter counting problem is itself trivially #P-complete by parsimonious reduction from SAT. Because only the most signcant figues are relevat, the reduction also holds for approximate counting. Similarly robust reductions exist for the

forR.

other relations listed above. The reductions employed in many proofs of #P-hardness, however, appeal to less trivial arithmetic such as polyno-

mial interpolation (see, e.g., (97J) and consequently say nothing about the

complexity of approximate counting. D We conclude our introductory material by quoting some results on the complexity of approximate counting and generation problems for p-relations as a class. From our earlier discussion, it might be expected that these

problems are in general intermediate in difculty between exact counting 5The minimum vertex degree must be at least. en, where n is the number of vertices and 0 0( co( 1/2 is fied. Note that the corresponding existence problem is in P, and that the counting problem is solvable exactly in polynomial time if c:; 1/2 (31).

Proof: Observe that the generic p-relation NCOMP defined by

NCOMP(M,x,w) = tY: M encodes an NP-machine and wy an accepting computation of M on input x J is self-reducible, and apply Theorem 1.18 and the reduction of Theorem 1.10. D Note that the class of (exactly) uniform generation problems for p-

relations lies within the third level of the hierarchy. By contrast, recal that

the corresponding class #P of exact counting problems is not known to lie within any fied level of the hierarchy.

2.1 THE MARKOV CHAIN APPROACH

43

2.1 The Markov chain approach to Chapter 2

i

generation problems Consider the following dynamic stochastic technique for generating objects

I

Markov chains and rapid mixing

unormly at random from a finite set S, assumed very large. Suppose it is possible to construct a Markov chain whose states may be identified with the elements of S. Suppose alo that the chain is ergodic with uniform stationary distribution; in other words, if the chain is alowed to evolve for t

steps the distribution of the final state approaches the uniform distribution as t - 00, irrespective of initial state. Then an alost uniform generation procedure for S is obtained by simulating the Markov chain for sufciently many steps, from some arbitrary initial state, and outputting the element of S corresponding to the final state.

This monograph is primarily concerned with positive results about counting and generating structures described by natural p-relations. With the exception of certain relations of unary type, where there is a considerable body of work on analytic (Le., closed form) counting estimates, such results

In the following chapters we shall exploit this approach to obtain effcient almost uniform generators for several natural relations as discussed in Chapter 1. More specificaly, if R is a relation and x a problem instance

are rare in the kinds of interesting cases identified in Section 1.5. Ideally,

with non-empty solution set R(x), the idea is to construct a Markov chain

we would like to have avalable some algorithmic paradigms with reason-

MC(x) some or al of whose states correspond to the structures in R(x).

ably wide applicabilty. To this end, we investigate here a very general

Transitions in the chain will correspond to simple local perturbations of the

approach to generation problems. The approach is based on simulating a

structures themselves. This technique was suggested by Andrei Broder ¡16)

;¡imple dynamic stochastic process, namely a finite Markov chain, which

'lost always be self-reducible, the results we obtain carry over directly

as a means of generating perfect matchings in dense bipartite graphs: in Chapter 3 we will look at this problem in detail and show for the first time that the technique works. The same approach can be used to sample the structures from more general probabilty distributions by adjusting

GO the corresponding counting problems by virtue of the observations of

the stationary distribution of the Markov chain accordingly. Problems of

,ection 1.4.

this nature arise frequently in Monte Carlo investigations of physical sys-

moves around a space containing the structures of interest and converges GO some desired distribution on them. Since the relations we consider wil

tems ¡12J, where the states correspond to confgurations of the system and

Technques of this kind have been in use for some time, paricularly n the physical sciences; however, until recently the mathematical tools

tvailable for analysing the non-asymptotic behaviour of Markov chains were

tpparently not sensitive enough to provide usefu performance guarantees

'or the resulting algorithms. In this chapter, a method of analysis for

appropriate functions of the stationary process to physical constants or parameters. They alo lie at the heart of stochastic optimisation methods, such as simulated annealng ¡60), in which low cost confgurations are associated with large weights in the stationary distribution. We shal discuss these applications in more detail in Chapter 3.

\Æarkov chains is developed which is related to recent work in graph theory m the connection between the subdominant eigenvalues of a graph and its ixpansion properties. As well as being of interest in its own right, this wil mabIe us in later chapters to demonstrate for the first time the existence of iffcient approximation algorithms for a number of important counting and

Assuming that the local structure of the Markov chain is simple, so that individual transitions can be simulated at smal cost, the effciency

~eneration problems. It wil also aford a deeper insight into the nature of

some polynomial p. Since the number of states is in general exponentialy

;he approximations themselves.

large, rapid mixng demands, informaly, that the chain should lose its memory afer visiting only a small fraction of its state space. In applying

of the above procedure depends crucially on the rate of convergence of

the chain. Specifically, the chain MC(x) should be rapidly mixing in the sense that it is close to stationarity after only p(lxl) simulation steps, for

2.1 THE MARKOV CHAIN APPROACH

2.1 THE MARKOV CHAIN APPROACH

14

;his technique, we are therefore faced with the problem of deriving a priori )ounds on the number of simulation steps requied to achieve a distribution which is sufciently close to the limit for the purpose at hand.

In this case, we have that 7r(t)' = 7r(O)' pt _ 7r' pointwise as t _ 00,

i

and the limit is independent of 7r(O)'. The stationary distribution 7r' is the unique vector satisfyng 7r' P = 7r', L:i 1Ii = 1, i.e., the unique normalsed left eigenvector of P with eigenvalue 1. Necessar and sufcient conditions

i

for ergodicity are that the chain should be (a) irrducible, i.e., for each

In the traditional theory of Markov chains, the question of rate of conV'ergence has received relatively little attention. In recent years, however,

it has emerged as an important research area, and a number of analytic Gechnques have been explored by vaious authors. These include Fourier

malysis and group representation theory (23), and stochastic approaches

45

pair of states i,j E (N), there is an s E N such that p~;) ~ 0 (j can be reached from i in a finite number of steps); and (b) aperiodic, Le.,

gcdfs :p~;) ~ OJ = 1 for al i,j E (N).

!Uch as coupling (1) and stopping times (3). Although these methods are

Consider now the problem discussed earlier of sampling elements of the

~legant and yield tight bounds for simple chains which possess a highly iymmetrical structure, the analysis involved appears to become extremely iiffcult in more complex examples. In this monograph we develop a simple ~haracterisation of rapid mixg in terms of a structural property of the l1derlying graph of the chain. The characterisation is based on the clasiical relationship between rate of convergence and the eigenvaues of the

state space, assumed very large, according to the stationary distribution 7r' .

The desired distribution can be realsed by picking an arbitrar initial state and simulating the transitions of the Markov chain according to the prob-

abilties Pij, which we assume can be computed localy as requied. As

;ransition matrix. As we shal see later, it turns out to be a powerfu tool

the number t of simulation steps increases, the distribution of the random vaiable Xt wil approach 7r'. In order to investigate the rate of approach to stationarity, we define the following time-dependent measure of deviation

:Or obtaining usefu analytic bounds for a number of interesting chains.

from the limit: for any non-empty subset U ç (N), the relative pointwise distance (r.p.d.) over U afer t steps is given by

(n our development, we wi assume that the reader has a passing familarity

with the elementar theory of finite Markov chains in discrete time: a

I (t) I

ßu(t) = 0,3EU ~ax Pij1Ij - 1Ij

ietaied introduction can be found in, for example, (34, Chapter XV). In ;he remainder of this section, we establish some terminology and notation tnd quote without proof some basic facts.

Thus ßu(t) is just the largest relative diference between 7r(t)' and 7r' at any state j E U, maxmised over al possible initial states i E U. i The inclusion

Let the sequence of random vaiables (Xt):o be a time-homogeneous \1arkov chain on a finite state space (N) = fO, 1, . . . , N - 1 J, N ~ 1, with

of the parameter U merely alows us to specify that certain portions of the state space are not relevat in the sampling process, as wi prove helpfu

¡ransition matri P = (Pij)i:~io' (Unless otherwse stated, all Markov

0,3-

~hains in this monograph wi be assumed to be of this form.) Thus for tny ordered pair i,j of states the quantity Pij = Pr (Xt+! = j I Xt = i) is

later. In the case that U = (N), we shal omit the subscript and write

simply ß in place of ßlNJ' The aim of this chapter is to obtain usefu bounds on ßu as a fuction of t. In paricular, we want to investigate

;he transition probability from state i to state j and is independent of t. rhe matrix P is non-negative and stochastic, i.e., its row sums are al unty. ¡¡or s E N, the s-step transition matrix is simply the power ps = (p~;)); thus

conditions under which the chains MC(x) are rapidly mixng in the sense

7~;) = Pr(Xt+s = j I Xt = i), independent of t. We denote the distribution

hence both) of the following equivaent conditions holds:

)f Xt by the row vector 7r(t)' = (7I~t)):~i. so that 1I~t) = Pr(Xt = i). flere 7r(O)' denotes the initial distribution, and 7r(t)' = 7r(O)' pt for al tEN. Usualy we wi have 1I~O) = 1 for some i E (N) (and 0 elsewhere); i is then ~aled the initial state. The chain is ergodic if there exists a distribution 7r' = (1Iö) ~ 0 over (N) !Uch that

li m (s) p..

s-+oo °3

=1Ij

V i,j E (N).

that ß(t) becomes very small in polynomial time.

An ergodic Markov chain is said to be (time-)reversible iff either (and

(trl) For all i,j E (N), Pij 1Ii = Pji 1Ij.

lWe have chosen this measure by analogy with our definition of almost uniform

generation in Chapter 1. Other measures, such as the variation distance, are also possible. For most interesting chains, this choice makes no essential difference to the rapid mixng criterion we are about to develop. We shal return to this point towards the end of this chapter.

6

2.2 CONDUCTANCE AND THE RATE OF CONVERGENCE (tr2) The matrix Dl/2 PD-l/2 is symmetric, where Dl/2 is the diagonal

mat.rixd'iag(1/2 110 ,...,1IN-l an is its.inverse. 1/2) d D-l/2' .

2.2 CONDUCTANCE AND THE RATE OF CONVERGENCE

47

As we have already stated, the stationar distribution 7r' of an ergodic chain is a left eigenvector of P with associated eigenvalue Ào = 1. Let lÀi : 1 ~ i ~ N - 1), with Ài E C, be the remaining eigenvaues (not

Condition (trl) says that, in the stationar distribution, the expected numbers of transitions per unt time from state i to state j and from state j to state i are equal, and is often referred to as the "detaied balance" property. It is easy to verify that, for any ergodic chain, if 7r' is any positive vector satisfyng (trl) and the normalzation condition Ei 1Ii = 1, then the chain is reversible and 7r' is its stationary distribution.

necessarily distinct) of P. By standard Perron- Frobenius theory for nonnegative matrices ¡ 85), these satisfy I Ài I -c 1 for 1 ~ i ~ N - 1. Futhermore,

the transient behaviour of the chain, and hence its rate of convergence, is governed by the magntude of the eigenvaues Ài' In the reversible case, condition (tr2) of the definition implies that the eigenvaues of P are just those of the similar symmetric matrix Dl/2pD-l/2, and so are al reaL.

This fact leads to a clean formulation of the above dependence, expressed

In our development we shal work exclusively with reversible chains. As we shal see, this assumption simplifies the analysis of the time-dependent

in the following pair of propositions.

behaviour. Moreover, since most chains arising in the computational applications of interest to us have this property, it does not represent a sig-

Proposition 2.1 Let P be the transition matrix of an ergodic reversible

nicant restriction. For a brief discussion of non-reversible chains, see the

Appendix.

Markov chain, 7r' its stationary distribution and Pi : 0 ~ i ~ N - 1)

its (necessarily real) eigenvalues, with Ào = 1. Then for any non-empty subset U ç ¡N) and all tEN, the relative pointwise distance ~u(t) satisfies

U _' ,

It is iluminating to identify an ergodic reversible chain with a weighted undiected graph (containing self-loops) as follows. The vertex set of the

~ (t) -c À~ax min1li iEU

!7aph is the state space ¡N) of the chain, and for each pair of states i, j (which need not be distinct) the edge (i,j) has weight Wij = 1IiPij = 1IjPji.

By detaied balance, this definition is consistent. Thus there is an edge of non-zero weight between i and j iff Pij :; O. We cal this graph the underlying graph of the chain. It should be clear that such a chain is

uniquely specified by its underlying graph.

where Àmax = maxl IÀil : 1 ~ i ~ N - 1).

Proof: Let Dl/2 and D-l/2 be as in the definition of reversibilty, so that the matrix A = Dl/2PD-l/2 is symmetric with the same eigenvaues as P, and these are real. This implies that we can select an orthonormal basis le(i)' : 0 ~ i ~ N - 1) for )RN consisting of left eigenvectors of A, where

2.2 Conductance and the rate of convergence

e(i)' = (e;i)) has associated eigenvaue Ài and e;O) = 11//2 for j E ¡N). Following ¡59), A has the spectral representation

N-l N-l A = ¿ Àie(i)e(i)' = ¿ ÀiE(i) , i=O i=O

(n this section, we establish an intimate relationship between the rate of convergence of an ergodic reversible chain and a certain structural property,

caled the conductance, of its underlying graph. Essentialy, such a chain will turn out to converge fast if and only if the conductance is not too smal.

where E(i) = eU)eU)' is a dyad (Le., has rank 1) with E(i) E(j = 0 for

The crucial step in the proof is a connection between the conductance and the second eigenvaue of the transition matrix of the chain. Similar rela" tionships between sub dominant eigenvalues of a graph and a more familar 5tructural property, known as the expansion or magnification, have been established in a different context by Alon ¡4) and Alon and Milan ¡5), and

hence

their relevance for the rate of convergence of certain Markov chains noted by Aldous ¡2).

i =I j, and E(i)2 = E(i). It follows that, for any tEN, At = Ei À!E(i), and

N-l pt = D-l/2 At Dl/2 = ¿ À!(D-l/2é))(e(i)' Dl/2) i=O

N-l = IN 7r' + ¿ À!(D-l/2é))(e(i)' Dl/2),

i=l

48

2.2 CONDUCTANCE AND THE RATE OF CONVERGENCE

where IN is the N-vector al of whose entries are 1; in component form, N-1 V 7rj i=1

By definition, the r.p.d. du is therefore given by

L"i=1\ teÇi) ". 3e(i) k I I"N-:1 -- max .."' "7rk 3,"kEU "3

49

in the sense indicated earlier if and only if ),max is suitably bounded away from 1. The first of these conditions can be checked immediately from our knowledge of Tr', and is rarely violated in practice. We therefore focus our attention on the second condition, which is not so easily handled. (Recal that P is assumed to be a large matrix, so that direct numerical evauation

PJ2 = trk + 1f:E ),~eJi)e~i) .

dU(t)

2.2 CONDUCTANCE AND THE RATE OF CONVERGENCE

of the eigenvalues is not feasible.) (2.1)

N~11 Çi)lle(i)1

max L" e3 k j,kEU i=1

-( ),~ax min 7r j

jEU -( ),~ax min 7r3"

Suppose the eigenvaues of P are ordered so that 1 = ),0 )- ),1 ~ .. . ~ ),N-1 )- -1. Then ),max = maxVi, I),N-11:¡ and the value of ),N-1 is signficant only if some of the eigenvalues are negative. Negative eigen-

values correspond to oscilatory, or "near-periodic" behaviour and cannot occur if each state is equipped with a sufciently la.ge self-loop probabilty. Specificaly, it is enough to have minj Pjj ~ 1/2. To see this, let IN denote

the N x N identity matrix and consider the non-negative matrix 2P - IN, whose eigenvaues are J.i = 2),i - 1. By Perron-Frobenius, J.i ~ -1 for all i E (N), which implies that ),N-1 ~ O.

jEU

In fact, negative eigenvalues never present an obstacle to rapid mixng where the second inequalty follows from the Cauchy-Schwarz inequalty

and the orthonormalty of the e(i)'. D

because any chain can be modified in a simple way so that the above con-

dition holds without risk of slowing down the convergence too much. All we need to do is increase the self-loop probabilty of every state by 1/2, as can easily be checked:

Proposition 2.2 With the notation and assumptions of Proposition 2.1,

the relative pointwise distance d(t) over (NJ satisfies

Proposition 2.3 With the notation of Proposition 2.1, suppose also that the eigenvalues of P are ordered so that 1 = ),0 )- ),1 ~ ... ~ ),N-1 )- -1.

d(t) ~ ),~ax

Then the modified chain with transition matrix P' = !(IN + P), where IN

for all even tEN. Moreover, if all eigenvalues of P are non-negative, the bound holds for al tEN.

is the N x N identity matrix, is also ergodic and reversible with the same stationary distribution, and its eigenvalues ~),a, similarly ordered, satisfy

),Ív-1 )- 0 and ),~ax = ),~ = !(1 + ),1). D

Proof: The equalty (2.1) in the proof of Proposition 2.1 stil holds. Set-

ting U = (NJ and k = j, we see that 2

eÇio)2 d(t) == d(N)(t) ~ max .=1. 3I~ ),~ax max -l IL~ -1 ),teçi)

jE(N) 7rj jE(Nj 7rj

for al (even) natural numbers t, where e(io)' is an eigenvector corresponding to an eigenvaue of modulus ),max' By orthonormalty of e(O)' and e(io)' and

the form of e(O)', this latter quantity is bounded below by ),~ax as rec¡uired. D

Remarks: (a) For the sake of simplicity, we have chosen the vaue 1/2 for the additional self-loop probabilty. A less crude approach would be to make this choice depend on the minimum diagonal element of P.

(b) Of course, in a practical simulation of the modifed chain, the waiting time at each state would be controlled by a separate random process. D

We turn now to the much more substantial problem of bounding the second eigenvalue ),1 away from 1. We shal do this by relating ),1 to a more accessible structural property of the underlying graph. Intuitively, we would expect an ergodic chain to converge rapidly if it is unkely to "get stuck" in any subset S of the state space whose total

Propositions 2.1 and 2.2 say that, provided Tr' is not extremely smal

in any state of interest, the convergence of a reversible chain wil be rapid

stationary probabilty is fairly smal. We can formalse this idea by considering the cut edges which separate S from the rest of the space in the

2.2 CONDUCTANCE AND THE RATE OF CONVERGENCE

,0

underlying graph, and stipulating that these must be capable of supporting a. sufciently large "flow" in the graph, viewed as a network. With this in mind, for any non-empty subset S of states with non-empty complement S

2.2 CONDUCTANCE AND THE RATE OF CONVERGENCE

Proof: Let e' = (ei);:~l be an eigenvector of P with associated eigen-

vaue À 0( 1, and define the matrix Q = IN - P (the "Laplace operator"

associated with P). Then clearly

In (N) we define the quantity (ls = Fs/Cs, where Cs = L7ri

the capacity of Si

iES

Fs = L Pij 7ri

the ergodic flow out of S.

51

e'Q = (1 - À)e'.

(2.2)

Define the subset of states S = ti E (N) : ei ;: Ol. Since P is stochastic and À 0( 1, it follows that Li ei = O. Hence 0 0( ISI 0( N, and we may assume without loss of generalty that Cs = LiES 7ri ~ 1/2. Now let ê' =

( êi) be the vector defined by

iES

jES

êi = f ei/7ri, 1. 0,

'lote that 0 0( Fs :: Cs 0( 1. (ls may be visualsed as the conditional

Jrobabilty that the stationary process crosses the cut from S to S in a iingle step, given that it stars in S. Finally, we define the conductance of ;he chain by

(l = min (l s . o-elsl-eN

059/2 ~t is easy to see that Fs = Fs for al such sets S. This implies that ¡'S = (ls(Cs/(1 - Cs)), so we may equivalently write

for i E Si otherwise.

Renumbering states as necessar, we shal assume that êo ~ êi ~ ... ~

êN-l, which implies alo that S = to, 1,..., r L for some r with 0 ~ r 0(

N - 1. Takng the inner product of (2.2) with ê' gives (2.3)

(e'Q, ê') = (1 - À)(e', ê').

The right-hand side of (2.3) is just

(l = min max t (l s, (lsl. o-elsl-eN

(2.4)

(1 - À) L 7ri ê~ .

iES

Now suppose that the chain is reversible, and let G be its underlying

Note that if Q = (aij) then qij = -Pij for i =l j, and qii = I-pii = L,#i Pij,

~aph. Then for al S as above we have

so we can expand the left-hand side of (2.3) as

Fs = Fs = L Wij , iES

jES

t fuction of the edge weights of G. The conductance (l == (l(G) may then )e viewed as a structural property of the weighted graph G. In view of the tbove remarks, we might hope that (l(G), which in some sense measures the ninimum relative connection strength between "small" subsets S and the 'est of the space, is related to the rate of convergence of the chain. This 'elationship is manfested via two separate bounds on the second eigen-

L L êiqjiej ~ L L êiQjiej iES jE(Nj iES jES - L L wijêiêj + L LWijê~ iES jES

#i

iES #i

2'" AA '" (.2 .2)

= - L. Wijeiej + L. Wij ei + ej

i-ej i-ej

L Wij(êi - êj)2 ,

(2.5)

i-ej

falue Ài.

~emma 2.4 For an ergodic reversible Markov chain with underlying graph

'J, the second eigenvalue Ài of the transition matrix satisfies Ài :: 1 _ (l(G)2

2

where the inequalty follows from the fact that al contributions with j ft S are positive. Using (2.4) and (2.5), equation (2.3) therefore yields 1- À ;: Li-ej wij(êi - êj)2

LiES 7riê~

(2.6)

2.2 CONDUCTANCE AND THE RATE OF CONVERGENCE

52

2.2 CONDUCTANCE AND THE RATE OF CONVERGENCE

53

This inequality ensures that the quotient in (2.7) is bounded below by ~(G),

Now consider the sum

so that finaly

i~j i~j iES

L wij(êi + êj)2 :S 2 L wij(ê~ + ê~) :S 2 L 7riê~ .

as required. 0

1 -.À ~ ~(G)2 2

Combining this with (2.6) gives

Combining Proposition 2.1 and Lemma 2.4, we arrive at the first major

1-.À ~ Ei~j "'0 wij(êi - êj)2 -"oeA2

Ei~j wij(êi + êj)2

result of this section, namely an upper bound on the distance from sta-

2 EiES 7riê~

tionarity of a reversible chain in terms of the conductance of its underlying graph.

L.iES "i i

'" A2 ' L.iES 7riei ~ ~2(Ei~jWij(êl -êJ))2

(2.7)

Theorem 2.5 Let G be the underlying graph of an ergodic reversible Markov chain all of whose eigenvalues are non-negative, and 7r' its station-

where we have used the Cauchy-Schwarz inequalty. To complete the proof,

ary distribution. Then for any non-empty subset U ç rN) and all tEN,

we need

the relative pointwise distance ßu(t) satisfies

to relate the quotient in (2.7) to the quantity ~(G).

ßu(t) :S. 0

To do this, consider the increasing sequence (Sk) ~=o of subsets of S with

( 1 _ ~(G)2 /2) t

Sk = to, . . . , k ì. The numerator of the quotient in (2.7) may be expressed

min 7r i

in terms of ergodic flows across the boundaries between successive sets Sk as follows:

iEU

Remarks: (a) In the interests of simplicity, Theorem 2.5 is stated only

~ Wij ei -A2) ej '" (A2 i~j

for chains with non-negative eigenvaues. Of course, Proposition 2.3 tell L Wij L (ê~ - ê~+1) i~j i'"k~j

us that any chain can be modied in a crude way so that this condition

holds: the effect of this operation on the conductance is to reduce it by a factor of 1/2. In practice it may often be possible to reason about negative

r

k=O iESk jf;Sk

L(ê~ - ê~+1) L Wij

eigenvaues on an ad hoc basis for the chai at hand. Proposition 2.1 and

Lemma 2.4 may then be used diectly to get a bound on ßu(t).

r

= L(ê~ - ê~+1)FSk .

(2.8)

k=O

(b) Theorem 2.5 says that ß(t) is bounded above by a fuction of the form exp(-(t - to)/ß), where ß and to are determined by ~(G) and 7r'.

Thus ß(t) is guaranteed to decrease at an exponential rate after an initial

Now the capacities of the Sk satisfy CSk :S Cs :S 1/2 for 0 :S k :S r, and

"delay" of to = 2~(G)-2In7r;;fn time unts, where 7rmin = mini7ri. Once

hence by definition of~, FSk 2: ~(G) Csk. We therefore get from (2.8)

this regime sets in, the actual exponential rate of convergence is governed

r

i~j k=Or

L wij(êl - ê~) 2: ~(G) L(ê~ - ê~+1)CSk

k

= ~(G) L(ê~ - ê~+1) L 7ri

k=O i=O r r

= ~(G) L 7ri L(ê~ - ê~+1)

i=O k=i

= ~(G) L 7riêl. iES

by the "time constant" ß = 2~( G) -2. The number of steps requied to ensure a r. p.d. of E is at most to + ß In E-l. We shal distil the essential features of this behaviour for our purposes into Corollar 2.8 of the next section. 0 Our next aim is to derive a parial converse of Theorem 2.5. First we

requie a bound on .Àl complementar to that of Lemma 2.4. Lemma 2.6 For an ergodic reversible Markov chain with underlying graph

G, the second eigenvalue .Àl of the transition matri satisfies .Àl 2: 1 - 2~(G).

54

2.2 CONDUCTANCE AND THE RATE OF CONVERGENCE

Proof: As in the proof of

Lemma 2.4, we work with the matrix Q = IN-P,

2.2 CONDUCTANCE AND THE RATE OF CONVERGENCE

55

Inequality (2.10) therefore yields

whose eigenvalues are fl - ),iÌ for ),i an eigenvaue of P. First we get a

general bound on ),1 using a vaiational principle.

1 - ),1 :: Fs Cs + 1 _ Cs :: 2 Cs = 2g)s .

Define B = D1/2QD-1/2; B is symmetric by vitue of reversibilty, and its eigenvaues are those of Q. Futhermore, if 7r' = (1Id is the stationary dis-

( 1 1) Fs

Since 8 was chosen arbitrarily, we have the bound

tribution of P, then the vector e' = (ei) with ei = 11//2 is a left eigenvector

),1~1-2g)(G),

of B with eigenvalue O. Now let g' = (gi) be any vector orthogonal to e'. Since B is symmetric with second smalest eigenvalue 1 - ),1, the classical

which completes the proof. 0

calculus of vaiations tells us that (2.9)

(1-),1)(g',g'):: (g'B,g').

To relate this to Q, introduce the vectors f' = (Ji) = g'D1/2 and f' = (Îi) = f'D-1. Then (2.9) becomes

(1 - ),1)(f', f') :: (f'Q, f'),

(2.10)

Proposition 2.2 and Lemma 2.6 together yield a converse of Theorem 2.5.

Theorem 2.7 Let G be the underlying graph of an ergodic reversible Markov chain all of whose eigenvalues are non-negative, and suppose that g)(G) :: 1/2. Then the relative pointwise distance ß(t) over (N) satisfies ß(t) ~ (1- 2g)(G))t

and this holds for al f' with ¿i Ji = O. The right-hand side of (2.10) can

be rewritten along similar lines to the proof of Lemma 2.4: it is a simple matter to check that

(f'Q, f') = L Wij(Îi - Îj)2 ,

i-lj

(2.11)

where as usual Wij = 1IiPij = 1IjPji denotes the weight of the edge (i, j) in G.

Now let 8 be any subset of states for which Cs :: 1/2. The idea is to select a particular vector f' for which (2.10) yields a good bound on g)s. With this in mind, set for i E 8;

~

fi = C~'-1Ii f 11'

,. \ , for i ~ 8.

Then clearly ¿i fi = 0, so (2.10) holds for f. Fuhermore, we have

A, f2i 11" i (I-CS)2 _ Cs _ L= 1Ii (+1 1) (f',f)=L;:=LC2+ l-Cs iE(NJ i iES S i~S

and from (2.11), since f' is constant on 8 and S,

for all t E fi.

o

Remarks: (a) For simplicity, and in paralel with Theorem 2.5, we have stated Theorem 2.7 only for chains with non-negative eigenvalues (and conductance at most 1/2). If the modification procedure of Proposition 2.3 is

applied to an arbitrary chain then both of these conditions are guaranteed to hold. For chains with negative eigenvaues (and g)(G) :: 1/2), the bound of Theorem 2.7 holds for all even t E fi, by Proposition 2.2.

(b) Lemma 2.4 and its proof parallel an earlier continuous result of Cheeger (19) for Riemannian manifolds. In the discrete setting, the lemma and its converse, Lemma 2.6, are closely related to recent work of Alon (4)

and Alon and Milman (5) (see also Dodziuk (27)) in which a relationship between a similar structural property of simple, unweighted graphs and the second eigenvaue of the adjacency matrix is established. This property, caled the "magnification" in (4, 5), measures the minimum number of vertices adjacent to a small subset 8 as a fraction of 181, and is a generalsation of the widely-studied concept of expansion for bipartite graphs. Our conductance g) is a weighted edge analogue of magnfication, and is the natural quantity to study in the present application.

(c) The signficance of Alon's result (4) as a sufcient condition for rapid

(f' Q, f) = L Wij C s + 1 - C s = Fs C s + 1 - C S iES

j~S

( 1 1)2 ( 1 1)2

convergence of certain Markov chains has been observed by vaious authors (see, e.g., (2, 17, 80)). In particular, Aldous (2) states a restricted form of Theorem 2.5 for random wals on reguar simple graphs. However,

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~

2.3 A CHARACTERISATION OF RAPID MIXING

58

Assume now that al chains in some family are reversible and are known to have non-negative eigenvalues; recal that the latter can always be ar-

ranged by boosting self-loop probabilties as in Proposition 2.3. Futher-

more, suppose that the minimum probabilty 7r~i~ assigned to any state in the stationary distribution of MC (x) satisfies Ig7r~(n -1 :: q'(lxl)

(2.12)

for al x E n and some polynomial q'. Then the results of the previous section give us the following elegant characterisation of the rapid mixng property.

Corollary 2.8 Let ~ MC(x) : x E n 1 be a family of ergodic reversible Markov chains, and let G(x) be the underlying graph of MC(x). Under the above assumptions, the family is rapidly mixing if and only if

2.3 A CHARACTERISATION OF RAPID MIXING

59

following example confirms that, for ergodic chains with negative eigenvalues, rapid mixng is not guaranteed by a polynomial lower bound on the conductance. Example 2.9 Consider the family of chains MC(n) parameterised on natural numbers n E N+, in which MC(n) has state space VI U V2 for disjoint Vl, V2 with !VI I = !V21 = N /2 and N = 2n. Define the matrix P = (Pij) by Pi. f 2/N if i E VI, j E V2 or i E V2, j E VI; J =1. 0 otherwise,

and let the transition matrix of MC(n) be (1 - a)P + aIN, where IN is the N x N identity matrix and the value of a wil be specified below. The chain MC(n) is obviously ergodic for 0 -( a -( 1; since it is symmetric, it is also reversible with uniform stationar distribution. To calculate (l(G(n)), consider any non-empty subset S = Au B of states,

with A ç VI, B ç V2 and ISI :: N/2. Writing a = IAI/N and b = IBI/N,

(l(G(x)) ;: l/p(lxl)

we have Cs = a + band

for all x E n and some polynomial p. Fs = 2(1 ~ a) (a(N/2 - bN) + b(N/2 - aN)).

Proof: We may reformulate the results of the previous section in terms of r(x) as follows:

1- 2(l( G x( InE-1:: ~tr.,(_\\?(x) InE +ln7rmin ))r(x)(E):: 2 (-1 -1).

It follows that

Fs (4ab ) Cs a + b

(ls == - = (1 - a) 1 - - .

(2.13)

The latter quantity is minimised under the constraint a + b :: 1/2 at the

The upper bound on r(x) is a direct consequence of Theorem 2.5. For the lower bound, note from Proposition 2.2 that r(x)(E) ;: (Inc1)/(ln-A;~) :; (In E-1 )-Amax/(I- -Amax), using the fact that In Z-l -( (1- z)/ z for z E (0,1),

values a = b = 1/4, whence (l(G(n)) = min (ls = (1 - a)/2.

s

(2.14)

and then apply Lemma 2.6.

Now consider the probabilty that the chain, when started from some initial

The corollary follows immediately from (2.13). 0

state in VI, will be found in VI at time t = 2m + 1 for mEN. This can happen only if at least one self-loop is traversed during this period, so the

Corollary 2.8 provides a

partial characterisation of the rapid mixng

property in terms of the conductance of the underlying graphs of the family.

The characterisation is incomplete because of the non-negativity assumption on the eigenvalues and the lower bound (2.12) imposed on 7r~i~' We

can justify the restriction to chains having non-negative eigenvaues on the grounds that our ultimate aim is to construct effcient sampling proced~es for a given family of distributions. The modification procedure of Proposition 2.3 is effective and cannot destroy the rapid mixng property of the original family: it is therefore reasonable to incorporate it into the chains we consider, and we wil always do this. For the sake of completeness, the

probabilty is at most 1 - (1 - at If we set a = N-1 (say), then if tis permitted to grow only polynomialy with n this probabilty tends to 0 as n -- 00. It follows that the family is not rapidly mixng. Equalty (2.14), however, ensures that the conductance is large. Note that the above chains fail to converge fast because they are "alost

periodic". P is the transition matrix of the random walk on the complete bipartite graph with vertex set VI U 1I, which is periodic with period 2. MC(n) is obtained from P by adding a self-loop probabilty a to each state. The effect is to increase the smalest eigenvaue -AN-1 from -1 to -1 + 2a, and the second eigenvaue -A1 from 0 to a. For a = N-1, we therefore have

60

2.3 A CHARACTERISATION OF RAPID MIXING

2.3 A CHARACTERISATION OF RAPID MIXING

61

.Àmax = 1- 2N-l. On the other hand, taking 0: = 1/2 corresponds precisely to the modification procedure of Proposition 2.3. In this case, of course,

distribution at every point; and secondly it demands that convergence be rapid from every initial state. We can formulate a less strict definition as

.Àmax = 1/2 and the chains become rapidly mixng. D

follows. Define the variation distance from initial state i by

Next we consider the effect of the stationary distribution on the char-

~rar(t) = l L IpW -1ljl,

j

acterisation of Corollary 2.8. The following example demonstrates that the

assumption of a lower bound of the form (2.12) on 1l~i~ is necessar.

and let Trr(€) = minit EN: ~rar(t') ~ € for al t' ? tJ-. By relating

Example 2.10 For n E N+, let MC(n) be a one-dimensional geomet-

bounds on TVarmay be obtained (see, e.g., (871):

ric random wal on state space (N), where N = 2n, with transition

~ir(t) to .Àmax in similar fashion to Propositions 2.1 and 2.2, the following

probabilties Pi(iH) = 1/3, Pi(i-l) = 2/3 for 1 ~ i ~ N - 2, and (i) Trr(€) ~ (1- .Àmax)-l (ln1l;l + lnel);

POD = P(N-l)(N-l) = 2/3, POl = P(N-l)(N-2) = 1/3. (Thus states 0 and

N -1 form reflecting barriers.) Then MC(n) is ergodic and reversible with

stationary distribution 1li = 2-i-l for 0 ~ i ~ N - 2 and 1l(N-l) = 2-NH. FUrthermore, the conductance cp(G(n)) is exactly 1/3, as may readily be verified. Hence, if the chain is modified as in Proposition 2.3 to eliminate

negative eigenvalues, the conductance is stil 1/6. However, this family is not rapidly mixng since the number of states reachable from any initial state in t steps is at most 2t + 1. This does not contradict Corollar 2.8

since Ig1l;:~ = _2n + 1. D Fortunately, pathological cases such as the one above rarely occur in

our applications. Moreover, states with extremely smal weight in the stationary distribution are typically not relevant to the sampling process, and their effect can be eliminated by working with ~u(t) for some suitably chosen subset U of states. Hence for most practical purposes the conductance may be taken as a reliable characterisation of rapid mixng for familes of reversible Markov chais.

In the remainder of this monograph, extensive use wil be made of the positive part of the above characterisation to show that certain natural familes are rapidly mixng, thus providing effcient sampling schemes for

the associated stationar distributions. We shal see that, for several chains with a rather complex structure, the conductance may be quite accessible while the rate of convergence is apparently not easily investigated by existing methods. Thus we contend that the characterisation of rapid mixng presented in this chapter is a potentialy powerfu tool for analysing the time-dependent behaviour of a much wider class of Markov chains than has

hitherto been possible. '-.

(ii) m~ Trr(€) ? l.Àmax(1- .Àmax)-lln(2€)-1.

i

Using the bounds on .Àl from Lemmas 2.4 and 2.6, we therefore get upper and lower bounds on Tvar in terms of the conductance cp that are very similar to those of (2.13) in the proof of Corollar 2.8. Thus we see that cp also characterises the rapid mixng property when formulated in terms of variation distance. Note, however, that in this formulation the initial state plays a signficant role. In paricular, the upper bound (i) depends only on the stationary probabilty of the initial state, rather than on 1lmin'

This is of practical vaue in cases where the stationary distribution contains states of very smal probabilty (so 1lmin is tiny) but we are able to choose

an initial state whose stationary probabilty is quite large. On the other hand, the lower bound (ü) does not rule out the possibilty that the chain converges fast in vaiation distance from certain initial states even when the conductance is smal; such examples exist. We could in fact have developed the theory in this chapter, and in much of the rest of the monograph, in terms of variation distance. Indeed, for many puroses, including the reduction from counting to generation of

Section 1.4, it is enough to generate structures from a distribution whose vaiation distance from the stationar distribution is smal. However, we have chosen to work with r.p.d. here for consistency with our rather strict

definition of alost unform generation in Chapter 1. It should be clear from the foregoing remarks that this choice makes little essential difference to our main results. D

We close this chapter with an observtion which is frequently useful

Remark: Our definition of rapid mixng is rather a strict one because it is

in applications. Many natural Markov chains can be viewed as a simple

based on the r.p.d., which is a very severe measure for two reasons. Firstly,

random wal on a graph H = (V, E) in which transitions are made from

it demands that the distribution of the chain be close to the stationary

any vertex v to an adjacent vertex with probabilty ß / d, where d is the

62

2.3 A CHARACTERISATION OF RAPID MIXING

maxmum vertex degree of Hand ß ~ 1 is a positive constant. In addition, v has self-loop probabilty 1 - ß deg( v) / d, where deg( v) is the degree of v

in H. If H is connected and ß -: 1 then the chain is certainly ergodic;

Chapter 3

symmetry ensures that it is reversible with uniform stationary distribution. Note that in the underlying graph al edges of H have equal non-zero weight, while all remaining edges other than self-loops have weight zero.

For any subset S ç V, let r(S) denote the cut set in H defined by S,

Direct Applications

i.e., the set of edges in E with one endpoint in S and one endpoint in V - S, and define the (edge) magnification ¡i(H) of H by

(H) = min Ir(S)1

¡i O-:ISI::IVI/2 ISI

Clearly, 0 -: ¡i(H) ~ d. The following equivaence is immediate from the definition of conductance. Proposition 2.11 Let G be the underlying graph of an ergodic random

walk on a graph H with maximum degree d and transition probabilities ß / d between distinct adjacent states. Then the conductance of G is given by

We have seen in Chapter 2 that the Markov chain simulation paradigm

provides an elegant general approach to generation problems, and have developed some theoretical machinery for analysing the effciency of the resulting algorithms. The purpose of this chapter is to demonstrate the utilty of the approach by applying it to some concrete and non-trivial examples. We shall show how to generate various combinatorial structures by constructing suitable ergodic Markov chains having the structures as states and transitions corresponding to simple local perturbations of the

(l(G) = ß¡i(H)/d. 0

structures. The rate of convergence wil be investigated using the tech-

niques of Chapter 2, and in particular the rapid mixng characterisation

In a typical family of random wals MC (x), the degree d is fairly smal (i.e., bounded by a polynomial in Ix!), so the rapid mixng criterion boils lower bounds on the magncation. This view down to finding polynomial

of Corollary 2.8. In each case, the detailed structure of the Markov chain wil enable us to estimate the conductance of its underlying graph, and we develop a useful general methodology for doing this. Our results constitute

simplies the analysis of familes of this kind. As the following example

apparently the first demonstrations of rapid mixng for Markov chains with

ilustrates, however, such a bound on the magnfication is of no signficance

genuinely complex structure. As corollaries, we deduce the existence of ef-

when the degree is large.

ficient approximation algorithms for two signficant #P-complete counting problems.

Example 2.12 For n E N+, define the graph H(n) as follows. Let Hi, H2 be two copies of the complete graph KN/2, where N = 2n. Then H(n) the disjoint union of Hi and H2 together with a perfect matching consists of between vertices of H i and vertices of H 2. Let MC (n) be a random wal

3.1 Some simple examples

on H(n) as above. Such a chain is clearly ergodic, and it is not hard to verify that ¡i(H(n)) = 1, with the minimum vaue of r(S)/lsl attained

Before tacklng some more substantial examples, let us first apply the techniques of Chapter 2 to construct natural Markov chain generators for a

when S is the vertex set of one of the Hi. However, the degree of H(n) is

few very simple structures. The generation problems considered in this section are not particularly interesting from a computational point of view

N/2 = 2n-i, which precludes the possibilty of rapid mixng regardless of the value of ß. An obstacle to rapid convergence is presented by the edges

between the vertex sets of Hi and H2, which constitute a constriction to flow in the underlying graph. 0

as a number of effcient exact methods exist for their solution. Moreover,

the associated counting problems are trivial. However, our analysis wil serve to ilustrate what is involved in practical applications of the rapid mixng characterisation of Chapter 2. It will also allow us to develop some additional technology which wil play a central rôle in later proofs.

64

3.1 SOME SIMPLE EXAMPLES

Consider first the relation B which associates with each natural number n the set B(n) = to,1)n of bit vectors of length n. We proceeed to

construct a family of Markov chains MC(n) which can be used as an al-

3.1 SOME SIMPLE EXAMPLES

65

Remark: Inspection of the proof of Corollary 2.8 reveals that the actual number of simulation steps performed by the generator on input (n, €) is O(n2(n + 1ge1)). D

most uniform generator for B following the paradigm of Figue 2.1. The

most natural process to look at here is one which moves around the state space B(n) by flipping a single random bit on each transition. Thus we can view MC(n) as a random wal on the n-reguar graph H(n) with vertex set B(n) and edge set

We turn now to the proof of Theorem 3.1, which ilustrates a technique which wi be employed throughout this chapter.

t(u,v) E B(n) x B(n) : Ð(u,v) = 1),

to specify a canonical simple path in H(n) between each ordered pair of

Proof of Theorem 3.1: Let N = 2n be the number of states of MC(n). Our arguent hinges on the following observtion. Suppose it is possible

where Ð denotes Hamming distance. Of course, H(n) is just the ndimensional hypercube. To avoid problems of periodicity, we add a self-

loop probabilty of 1/2 to each state (Le., ß = 1/2 in the terminology of Proposition 2.11); note that this also dispenses with the problem of nega-

distinct states in such a way that no oriented edge of H ( n) is contained in

more than bN of the paths. If S is any subset of states with 0 -c ISI :5 N/2, then the number of paths which cross the cut from S to S is clearly

ISI(N - ISI) ~ ISIN/2.

tive eigenvalues as in Proposition 2.3. This gives us an ergodic reversible

Markov chain with uniform stationary distribution. Tuning now to the question of effciency, it is clear that conditions (mc1)-(mc3) of Section 2.3 hold in this case: we may select on as initial state and simulate individual steps in O(logn) time on an OCM (see

Proposition 1.3). The effciency of the generation procedure is therefore governed by the rate of convergence of the chain. The results of Section 2.3

in turn imply that this depends on the magncation of H(n). Fortunately, a suitable bound on this quantity is not too hard to come by.

Theorem 3.1 The magnification of the n-dimensional hypercube satisfies

¡i(H(n)) ~ 1. Before proving Theorem 3.1, let us fist confrm that the resulting generation procedure is effcient.

Hence for any such S the number of cut edges Ir(S)1 is bounded below by

ISIN /2bN = ISI/2b, whence

. Ir(S)1 1

¡i(H(n)) = mJn -- ~ 2b'

(3.1)

The problem of bounding ¡i(H(n)) below can therefore be reduced to one of defining a collection of canonical paths in H(n) which are "suffciently edge disjoint", as measured by the bottleneck parameter b.

We now proceed to define a suitable set of paths. Let u = (Ui)~:Ol and v = (Vi)~:Ol be distinct elements of B(n), and ii -c ... -c i¡ be the positions

in which u and v dier. Then for 1 :5 j :5 l, the jth edge of the canonical path from u to v corresponds to a transition in which the ijth bit is flpped from Uij to Vij'

Consider now an arbitrar tranition t of MC(n) (or, equivaently, an oriented edge of H(n)); our aim is to bound the number of paths which con-

Corollary 3.2 Simulation of the above family of Markov chains yields a

tain t. Suppose that t takes state W = (Wi) to state W' = (wD by flpping

J.P. almost uniform generator for the bit vector relation B.

the vaue of Wk, and let P( t) denote the set of paths containing t, viewed as ordered pais of states. Rather than counting elements of pet) directly, we

wi set up an injective mapping from pet) into the state space B(n); this Proof: By the preceding discussion, it is enough to check the rapid IIixing condition (mc4). Theorem 3.1 and Proposition 2.11 imply that the conductance of the underlying graph of MC(n) is bounded below by 1/2n. Clearly, the minimum stationar probabilty satisfies 19 7l:~ = -no Since the conditions of Corollary 2.8 are satisfied, we conclude that the family of Markov chains is rapidly mixng as required. D

wil yield an upper bound on the bottleneck parameter b appearing in (3.1).

The mapping at: pet) -- B(n) is defined as follows: given an ordered pai (u,v) E pet), set at(u,v) = (sï), where

Sit Ui, =k -c.ii :5-c k;n. Vi, 0:5

3.1 SOME SIMPLE EXAMPLES

66

3.1 SOME SIMPLE EXAMPLES

67

Thus O"t (u, v) agrees with U on the first k + 1 bits and with v on the

what the injective mapping technique achieves. As we shall see presently,

remainder. Note that we can express this definition more succinctly as

it turns out to be rather generally applicable.

O"t(u, v) = u EB v EB w', where EB denotes bitwise exclusive-or.

the endpoints u and v,

We claim that O"t(u, v) is an unambiguous encoding of

so that O"t is indeed injective. To see this, simply note that Si, Ui = 1.f Wi,

o ~ i ~ k;

k -c i -c n;

Vi = f w~, 0 ~ ~ ~ k;

1. Si, k -c i -c n.

Hence u and v may be recovered from knowledge of t and O"t(u, v), so O"t is

Other simple Markov chains may be analysed in a similar fashion. Examples of rapidly mixng familes include random walks on n-dimensional cubes of side d and a host of "card-shufHing" processes whose state space

is the set of permutations of n objects and whose transitions correspond to some natural shufing scheme (see, e.g., (231). Here we content ourselves with two fuher examples to ilustrate the generality of our approach before passing on to more interesting problems.

injective. It follows immediately that IP(t)1 ~ Nj in fact, since al vectors (Si) in the range of O"t satisfy Sk = Wk, we have the stronger result

IP(t)1 ~ IB(n)I/2 = N/2. Since t was chosen arbitrarily, the number of paths traversing any oriented edge cannot exceed N/2. Setting b = 1/2, inequalty (3.1) now yields the desired bound on the magnification ¡.(H(n)). 0

Example 3.3 Let us analyse a simple card-shufing process based on random transpositions, as studied in (24). For a natural number n, let Sn denote the set of permutations of the set (n) = to,..., n - I). Con-

sider a deck of n cards labelled with the elements of (n), and identify P = (Po, . . . , Pn-l) E Sn with the ordering of the deck in which the ith card from the top is Pi. We define a Markov chain MC(n) with state space Sn in which transitions are made by picking a pair of cards at random (without replacement) and interchanging them. As usual, we incorporate a self-loop

Remark: The bound of Theorem 3.1 is tight. To see this, let S be the subset of B(n) consisting of al vectors with first bit 0 and note that Ir(S)I/ISI = 1. Hence ¡.(H(n)) = 1 for the n-dimensional hypercube.

However, the bound on second eig~nvalue provided by Lemma 2.4, namely ).1 ~ 1 - 1/8n2, is not tight: the exact value of ).1 is 1 -c/n for a constant c.

o Some observations on the above proof are in order here. Passing from the conductance (a "cut" quantity) to paths (a "flow-lie" quantity) is analogous to transforming the problem into its dual: this is advantageous because we need only exhibit some good collection of paths rather than argue about all possible cuts. (This analogy is made explicit in the Appendix.) Similar ideas have been used before in the literature to investigate the connectivity properties of various graphs in other contexts (94). The principal

novelty of our proof lies in the use of the injective mapping technique to bound the number of paths which traverse an edge. This is not actualy necessary in this simple example as the paths could have been counted explicitly. The point is that in more complex cases the states of the chain will be less trivial structures, such as matchings in a graph, and we wîl have no useful information about their number - indeed, this is what we

wil ultimately be trying to compute. It is then crucial to be able to bound the maxmum number of paths through any edge in terms of the number of states without explicit knowledge of these quantities. This is precisely

probabilty of 1/2 for each state. Here again we have an ergodic random wal on a regular graph of degree n(n - 1)/2 and we need to look at its magnfication. A canonical path

between permutations u, v of the deck can be described as follows: for

successive vaues k = 0,. . ., n - 1, move the card Vk into position k (if it is not there already) by interchanging it with the current kth card. Consider now some transition t which interchanges the cards in positions j and k of

the permutation w, with j :; k, and as before let P(t) be the set of paths containing t. The injective mapping O"t : P(t) - Sn is a little more subtle here. For (u, v) E P(t), we refer to the positions of a given card in u, v,

W respectively as its initial, final and current positions. The permutation O"t(u,v) is then defined as follows:

(i) place the cards Wo,. .. , Wk-l in their initial positions; (ii) place the remaining n - k cards in the vacant positions in the order

in which they appear in the final permutation v. Let us now check that O"t is injective. Given t and O"t(u,v) we can uniquely

recover v as follows: the final positions of cards Wo,.. . , Wk-l are the same as their current positions, and the final order of the remaining cards may be read off from O"t (u, v). To recover u, note that the initial positions of

wo, . . . , Wk-l are just as in O"t( u, v); but these positions together determine,

68

3.1 SOME SIMPLE EXAMPLES

for every i, the current position of the card initially in position i, since each previous transition on the path involved moving one of Wo,..., Wk-1 into its final position. Hence we can deduce the initial positions of all cards, and so recover u.

Thus at is injective, and we have IP(t)1 :: N, where N = IBnl is the total number of states. It follows from (3.1), with b = 1, and Proposition 2.11

that the conductance of MC(n) is at least 1j2n(n - 1), so this famy of card-shufing processes is rapidly mixng. The number of simulation steps requied to achieve tolerance E is seen from the proof of Corollar 2.8 and an application of Stirling's approximation to be 0 (n4 (n 19 n + 19 c 1 )) . 0

Our final example ilustrates a simple but usefu generalsation of our path counting technque.

3.1 SOME SIMPLE EXAMPLES

69

P(t) denote the set of paths that contain edge t. We define the mapping at: P(t) ~ H(n,k) by at(u,v) = uEBvEBw', just as in the hypercube

example of Theorem 3.1. However, note that at is not necessarily injective here; what we must do is bound the expected number of paths with a given image under at. Note that at( U, v) EB w' = u EB v, so al paths with the same

image z have the same length m and the same probabilty m! -2 of being chosen, where 2m = IZ EB w'l. Moreover, the total number of paths which, given that they are chosen, have image z under at is

m-1 ~ m; 1( r!)2 2(m _ r _ I)! 2 = m(m _ I)! 2.

Example 3.4 We wil show how to construct an effcient Markov chain

(Here r corresponds to the distance along the path from u to w.) Thus the expected number of paths with image z is m!-2 m(m - I)! 2 = m-1. Finaly, summing over images, and noting that the range of at consists of

generator for subsets of an n-set of cardinality k. Equivaently, let SUBS

al subsets that contain j and do not contain i, we see that the expected

be the relation which associates with pais of natural numbers n, k all bit vectors of length n containg precisely k 1 's. For each pair n, k with o -: k -: n define the Markov chain MC(n, k) with state space sUBs(n, k)

number of paths containing t is

and transitions as follows: randomly select a pai of positions, one of which contains a 0 and the other a 1, and interchange their vaues. Adding a self-

loop probabilty of 1j2 to each state, we may view MC(n, k) as an ergodic random wal on a graph H(n, k) of degree k(n - k). This Markov chain corresponds essentialy to the classical Bernoul-Laplace diffsion modeL. It seems diffcult to

get a usefu bound on tt(H(n, k)) using canonical paths

as in the above examples. However, the following simple generalsation

of the path counting technque, suggested by Dagum et al (22J, does give a good bound. Suppose we define a set of paths (rather than a single

canonical path) between each pai of states in the underlying graph, and then choose one path from the set uniformly at random. If the sets of paths are defined in such a way that the expected number of paths containing any

given oriented edge is at most bN, where N is the number of states, then we can deduce that tt ;: 1j2b, exactly as before. (Clearly the same result holds if the paths are chosen from any probabilty distribution.) For our present example, let u, v E SUBS( n, k) and take as the set of patlis from u to v all shortest paths in H(n, k) from u to v; thus if lu EB vi = 2m,

there are m! 2 such paths, each of length m. (Each path corresponds to an ordering of the m elements of u - v and an ordering of the m elements of

min~k,n-k) -1 (k -1) (n - k -1)

L m m-1 m-1 m=l

N n

where N = (~) is the total number of subsets. Appealng to (3.1) with b = n-1 now yields tt(H(n, k)) ;: nj2. The conductance of MC(n, k) is

therefore at least nj4k(n - k), and we have rapid mixng. Note that the bound on second eigenvaue given by Lemma 2.4 is ).1 :: 1-n2 j32k2 (n- k)2,

which in the classical case n = 2k becomes ).1 :: 1 - 1j2n2. Again this is not tight: the exact vaue here is).l = 1 - cjn for a constant c. 0

The Markov chain familes mentioned in this section al possess a highly

symmetrical structure which makes them particularly easy to analyse. These and similar processes have alo been studied using other methods such as coupling, stopping times and group representation theory: see (1,3,23, 24J for a vaiety of examples. The time bounds obtained by these methods are generaly rather tighter than ours and can often be shown to be optimal. However, the fu power of our approach wil become apparent in the remainder of this monograph, where it wil permit the analysis of highly non-symmetric chains with only a little additional effort. Most sig-

v - u.) Now let t be a transition that takes state w to w' by interchanging

nificantly, such chains have so far not proved amenable to analysis by any

the vaues Wi = 0 and Wj = 1. For a paricular choice of random paths, let

of the other established methods.

70

3.2 APPROXIMATING THE PERMANENT

3.2 Approximating the permanent In this section we treat our first major example - the groundwork of Chap-

ter 2 wil begin to bear fruit in the form of a significant and unexpected approximabilty result. The permanent of an n x n matrix A with 0-1 entries aij is defined by

n-i per (A) = L II aio-(i) , 0- i=O where the sum is over all permutations a of

the set ¡n). Evaluating per(A) is

eq1Uvalent to counting perfect matchings (I-factors) in the bipartite graph

G = (Vi,V2,E), where Vi = '¡xo,...,xn-il, V2 = '¡Yo,...,Yn-il and ( Xi, Y j ) E E iff aij = 1. The permanent function arises naturaly in a number of fields, including algebra, combinatorial enumeration and the physical sciences, and has been an object of study by mathematicians since first appearing in 1812 in the work of Cauchy and Binet. We shall mention an application in statistical physics in the next section; for further background information see ¡n).

Despite considerable effort, and in sharp contrast with the syntacticaly very similar determinant, which can readily be evauated, no effcient procedure for computing the permanent is known. Convincing evidence for its inherent intractabilty was provided in the late 1970s by Valiant ¡96j, who

demonstrated that the problem of counting perfect matchings in a bipartite

graph is #P-complete. On the other hand, it is well known that the corre-

3.2 APPROXIMATING THE PERMANENT

71

in a large class of graphs, including al graphs which are suffciently dense. The existence of an effcient randomised approximation algorithm for the dense permanent is therefore established for the fist time. The crucial step is to show that the appropriate family of Markov chains on matchings is

rapidly mixng. In the next section, we wil present and analyse a more natural algorithm based on an alternative Markov chain.

Let G = (Vi, V2, E) be a bipartite graph with !Vil = !V21 = n, and for kEN let Mk(G) denote the set of matchings of size k in G. We assume throughout that G has a perfect matching, Le., that Mn(G) is non-empty. This assumption constitutes no real restriction since it can be tested in polynomial time as noted above. We view elements of E as unordered pais

of vertices, and matchings in G as subsets of E. If A, B ç E and e E E then A EE B denotes the symmetric difference of A and B, while A + e and

A - e denote the sets Au .¡ e Ì, A \ .¡ e Ì respectively.

Following Broder ¡16), we proceed to define a Markov chain MCpm (G)

with state space N = Mn(G) U Mn-i(G). Note that N includes auxliary states, namely "near-perfect" matchings in G, which wil permit free movement of the process between perfect matchings. Transitions in the chain are specified as follows: in any state MEN, choose an edge e = (u, v) E E

uniformly at random, and then

(i) if ME Mn(G) and e E M, move to state M' = M - e (Type 1 transition) ;

(ii) if ME Mn-i(G) and u,v are unmatched in M, move to M' = M + e (Type 2 transition);

sponding construction problem is solvable in polynomial time for arbitrar

graphs ¡30). The perfect matchings relation therefore belongs to the class discussed in Section 1.5, and the question of approximabilty is pertinent.

Until recently, little tangible progress had been made in the area of

(iii) if ME Mn-i(G), u is matched to win M, and v is unmatched in M, move to M' = (M + e) - (u,w) (Type 0 transition); (iv) in all other cases, do nothing.

effcient approximation algorithms for the permanent: to date, the best

known algorithm is due to Karmarkar et al ¡55) and has a runtime which grows exponentialy with the input size.i In 1985, Broder ¡16) proposed a Markov chain approach for almost unformly generating perfect matchings, which in view of the relationships of Section 1.4 could be used to count

For the sake of convenience, we introduce an additional self-loop probabilty of 1/2 for each state; Le., with probabilty 1/2 the process does not select a random edge as above but simply remains at M. MCpm(G) may be viewed as a random wal on an appropriate graph H of maxmum degree n. It is

them. However, Broder was unable to demonstrate that his algorithm is

not hard to see that H is connected, so MCpm (G) is ergodic and reversible

effcient. In this section, we fill this gap by showing that the method ~Òes indeed yield a f.p. randomised approximate counter for perfect matchings

with uniform stationary distribution.

lThe performance of this algorithm on an appropriate class of random instances, however, has recently been shown to be dramatically better (48J.

We now consider using the algorithm of Figure 2.1 in conjunction with the family of chains MCpm(G) as an almost uniform generator for perfect matchings. Stepwise simulation of transitions is readily performed by an OCM in time O(IEI), and we have aleady noted that the construction

72

3.2 APPROXIMATING THE PERMANENT

3.2 APPROXIMATING THE PERMANENT

73

problem for perfect matchings can be solved in polynomial time. Hence con-

Then, since each matching in lC(M) has at least n - 2 edges in common

ditions (mc1) and (mc2) hold. Condition (mc3), however, presents a prob-

with M, it is easy to see that IlC(M)1 S n2. Note that the sets lC(M)

lem, since G may in general contain many more near-perfect than perfect

partition N. This implies that INI S n2IMn(G)I, thus verifyng our earlier

matchings. Let us cal G dense if its minimum vertex degree is at least nj2.

claim that the near-perfect matchings are not too numerous. It is also

It is not hard to check (see below) that, if G is dense, IMn(G)ljINI 2: 1jn2, so that (mc3) holds. Remarkably, under this assumption it is alo possible

worth noting that this is the only point in the proof at which the biparite

to prove condition (mc4), i.e., that the family of Markov chains is rapidly

Next we define a canonical path in H between an ordered pair I, F of

mixng. This is a consequence of the following theorem.

perfect matchings (refer to Figue 3.1(a)). To do this, we first assume a

Theorem 3.5 For dense bipartite graphs G, the conductance of the underlying graph of the Markov chain MCpm(G) is at least 1j12n6.

Proof: It is sufcient to show that the graph H defining the random wal performed by MCpm (G) has magnfication

fied ordering of all even cycles of G, and distingush a start vertex in

each cycle. Now consider the symmetric difference I EB F; we may write this as a sequence C¡,..., Cr of disjoint even cycles, each of length at

least 4, where the indices respect the above ordering. The path from I to F involves unwinding each of the cycles Ci,..., Cr in turn in the following way. Suppose the cycle Ci has star vertex Uo and consists of the sequence

¡i(H) 2: 1j6n4,

(3.2)

for then the theorem follows from Proposition 2.11 with ßjd = 1j21EI 2:

1j2n2. To show (3.2) we proceed as in the proofs of the previous section by defining a set of canonical paths in H. If no transition occurs in more than blNI of these, (3.1) gives us a bound on the magnfication in terms of the bottleneck parameter b.

We begin by specifying, for each MEN, canonical paths to and from a unique closest perfect matching M E Mn (G) as follows, where u, v denote

the unmatched vertices (if any) of M:

(i) if ME Mn(G) then M = M and the path is empty;

(ii) if ME Mn-i(G) and (u,v) E E, then M = M +e and the path consists of a single Tye 2 transition; (ii) if M E Mn-i (G) and (u, v) ft E, fi some (u', v') E M such that (u,~'),(u',v) E E: note that at least one such edge must exist

by the density assumption on G. Then M = (M - (u',v')) + (u,v') + (u',v), and we specify one of

structure of G is used: we shal have more to say about this later.

the two possible paths of

length 2 from M to M, involving a Tye 0 transition followed by a Tye 2 transition. The canonical path from M to M consists of the same edges of H traversed " in the opposite direction. For future reference, we observe that no perfect matching is involved in too many canonical paths of the above form: for M E Mn(G), define the set

lC(M) = r M' EN: M' = MJ- .

of distinct vertices (uo, Vo, U¡, Vi, . . . , Ui, vi), where (Uj, Vj) E I for 0 S j S L and the remaining edges are in F. Then the fist step in the unwinding

of Ci is a Tye 1 transition which removes the edge (uo, vo). This is followed

by a sequence of L Tye 0 transitions, the jth of which replaces the edge (Uj,Vj) by (Uj,Vj_i). The unwinding is completed by a Tye 2 transition

which adds the edge (uo, vi).

The canonical path between any pai of matchings I, FEN is now defined as the concatenation of three segments as follows:

initial segment: follow the canonical path from I to I; main segment: follow the canonical path from I to F; final segment: follow the canonical path from F to F.

Now consider an arbitrary oriented edge of H, correspondig to a transition t in the Markov chain. We aim to establish an upper bound of the form blNI on the number of canonical paths which contain this transition. Suppose fist that t occurs in the initial segment of a path from I to F, where

I, FEN. Then it is clear from the definition of initial segment that the perfect matching I is unquely determined by t. But we have aleady seen that IlC(I)1 S n2. Since I E lC(I), the number of paths which contain t in

their initial segment is thus at most n21NI. A symmetrical argument shows that the number of paths containing t in their final segment is similarly bounded. To handle the main segments of the paths, we make use of the injective

mapping technique seen earlier. This wi obviate the need for any explicit

counting of structures in N, which is crucial here. Let t be a transition from M to M', where M,M' EN are distinct, and denote by P(t) the

74

3.2 APPROXIMATING THE PERMANENT

0-

0-

0-

0-

I Cl I ... I Ci-l I I Ci I I Ci+1 I

0-

0-

0-

0-

3.2 APPROXIMATING THE PERMANENT

75

0-

set of ordered pairs (I, F) of perfect matchings such that t is contained in the canonical path from I to F. We proceed to define, for each pair (I, F) E P(t), an encoding G't(I, F) E N from which I and F can be uniquely reconstructed. The intention is that, if Ci, . . . , Cr is the ordered

I Cr I

then the encoding should agree with I on Ci,..., Ci-i and on that portion

0-

sequence of cycles in I EB F, and t is traversed during the unwinding of Ci,

of Ci which has already been unwound, and with F elsewhere.

With this in mind, consider the set S = I EB F EB (M U M'). Since In F ç

start vertex of cycle Ci ~

0-

I0-I 0I0-I

/'. ¡"~ ¡~¡

/ ~ ~ /

~ / 0- 0-

/'.¡o'.¡~¡

/ ~ ~ /

MUM' ç I U F and III = IFI = 1M U M'I = n, elementary set theory

~/ 0/ 0-

tells us that ISI = n. Futhermore, suppose that some vertex u has degree greater than 1 in S, Le., we have (u,vi),(U,V2) E S for distinct vertices Vi, V2. Then necessarily (u, vi) and (u, V2) both lie in I EB F, which in turn

implies that neither edge lies in MUM'. Hence the vertex u must be unmatched in MUM'. From the form of the transitions, however, it is clear that MUM' contains at most one such vertex u = Uti moreover, this is the case iff t is a Tye 0 transition, and Ut must then be the start vertex of the cycle currently being unwound. In this case, we denote by el,t the edge of I incident with Ut.

We are now in a position to define the encoding: G't(I F) = f (I EB F EB (M U M')) - el,t, if t is Tye 0;

/ ~ ~ /

/ ~ / ~ / ~ ~ / ~ / ~ /

/ ~ ~ /

, 1. I EB F EB (M U M'), otherwise.

Figure 3.1(b) ilustrates this definition for a Type 0 transition. In view of

always a matching of cardinality at least the above discussion, G't(I,F) is n - 1, and hence an element of N. It remains for us to show that I and F can be recovered from it.

First observe that I EB F can be recovered immediately using the relation

Figure 3.1(a): A transition t on the canonical path from I to F

I EB F = f (G't(I, F) EB (M U M')) + el,t, if t is Tye 0;

1. (Tt(I, F) EB (M U M'), otherwise.

(Note that el,t is the unique edge that must be added to G't(I, F)EB(MUM') to ensure that I EB F is a union of disjoint cycles.) Thus we may infer the

0I Cl

0-

0I

... I Ci-l I

0-

/ Ci 0I ~ 0

/~ /~ ~ / ~ /' CHI

...

Figure 3.1(b): The corresponding encoding G't(I,F)

Cr

'.

ordered sequence Ci,. . . , Cr of cycles to be unwound on the path from I to F. The cycle Ci which is currently being unwound, together with its

parity with respect to I and F, is then determined by the transition t. The parity of all remaining cycles may be deduced from M and the cycle ordering. Finaly, the remaining portions of I and F may be recovered using the fact that In F = M \ (I EB F). Hence G't (I, F) uniquely determines the

pair (I, F), so G't is an injective mapping from P(t) to N. The existence of G't ensures that IP(t)1 S INI for any transition t. Since

also IK(M)I S n2 for any perfect matching M, we see that t is contained in

3.2 APPROXIMATING THE PERMANENT

76

¡he main segment of at most n41Nl paths. Combining this with the results

3.2 APPROXIMATING THE PERMANENT Let us now return to the approximation of

77

the permanent. Unfortu-

for initial and final segments derived earlier, we deduce that the maxmum

nately, the introduction of the density assumption means that we are no

;otal number of paths which contain t is bounded by

longer working with a self-reducible relation, so we cannot apply the reduction of Theorem 1.14 to get an approximate counter for perfect matchings. However, the same result is achieved by a more specialsed construction sketched by Broder (16J, which we spell out in detail below. A more natu-

(n2 + n2 + n4) INI $ 3n41NI. rakng b = 3n4 in (3.1) yields (3.2), which completes the proof. 0

ral reduction will be presented in the next section.

The characterisation of Corollar 2.8 now ensures that the Markov ~hains MCpm(G) constitute a rapidly mixng family. (Note that the num-

)er of perfect matchings in G is at most n!, so the minimum stationar )robabilty 7r;:~ of MCpm(G) satisfies 197r;:~ 2: -cnlgn for some con¡tant c.) In the light of the discussion preceding Theorem 3.5 we therefore

Corollary 3.7 There exists a J.P. randomised approximate counter for per-

fect matchings in dense bipartite graphs, and hence a f.p. randomised approximation scheme for the permanent of dense square 0-1 matrices. To simpli the proof, we fist derive an elementary statistical fact which

iave

says that, for a finite set S and a subset U ç S, the ratio IUI/ISI can be estimated effciently by alost unform sampling from S provided the ratio

:Jorollary 3.6 There exists a f.p. almost uniform generator for perfect

is not too smal.

natchings in dense bipartite graphs. 0

Proposition 3.8 Let S be a finite set and U a subset of S. Write p = ltemarks: (a) In his original paper (16), Broder claimed that the above

IUI/ISI. Suppose that t elements of S are selected independently and with

'apid mixng property holds under the same density assumption. However,

replacement from an almost uniform distribution within tolerance e E (0, 1), and let X denote the proportion of the sample which belong to U. Then for any 8 E (0,1), the sample size t required to ensure that

tis proof based on coupling ideas is both complex and fundamentaly Hawed.

lhe problem is that the "coupling" defined in (16) is not in fact a coupling

faithf copy of MCpm(G): ,his is explaied in detail by Mihai (74). As a result Broder has withdrawn tis proof (see Erratum to (16)). We see this as an indication that coupling )ecause one of

the two processes involved is not a

ind related methods may not be well suited to the analysis of Markov chains

Pr(X approximates p within ratio 1 + 5e) ~ 1 - 8

is at most (54/ep)ln(2/8).

"hich lack a high degree of symmetry.

b) A chain with a rather better conductance bound is obtained by modifiyng MCpm (G) slightly so that transitions are effected by selecting a random

rertex in V2 rather than a random edge. This gives us a random wal with ransition probabilties 1/2n rather than 1/21EI, and the same bound on

he magncation. c) In the case that G is the complete bipartite graph Kn,n, the Markov :hain in (b) may be viewed as a scheme for shufng a deck of n + 1 cards in vhich the top card is repeatedly interchanged with another card selected at

Proof: Writing p' for the expectation of X we have, since p' approximates p within ratio (1 + e)2 $ 1 + 3e,

Pr(X approximates p within ratio 1 + 5e)

2: Pr(X approximates p' within ratio 1 + e/2) 2: Pr(lX - p'l $ ep'13) 2: 1 - 2 exp( -ep't/27) ,

'.,

andom. Of course, MCpm(Kn,n) itself provides a generator for all permu-

ations of n objects, albeit rather indiectly. By appropriate choice of G, 'Ie can. alo generate in polynomial time vaious natural restricted classes .f permutations which satisfy the above density condition, such as displacenents or ménage arrangements. 0

where the last inequalty is derived from Chernoff's bound on tai of the

binomial distribution (6, Proposition 2.4). This latter expression certainly

exceeds 1 - 8 provided t 2: (54/ep) In(218). 0

3.2 APPROXIMATING THE PERMANENT

rs

)roof of Corollary 3.7: Let G = (Vi, V2, E) with !Vil = !V21 = n be

lense. For L = 0,..., n - 1, consider the graph Gi obtained from G by

3.2 APPROXIMATING THE PERMANENT

79

A similar bound holds for the proportion corresponding to (n - L - 1)-

matchings in G. Thus Pi and P2 are not too small. Putting ~ = €/20n and

,ppending L vertices to each part of the biparition, each new vertex being

Ó = I/Sn in Proposition 3.S now ensures that the sample size required to

onnected to al vertices of G in the opposite part. More precisely, Gi is the

achieve the specified accuracy is polynomially bounded in nand €-i. 0

~aph w;, V~, E'), where V; = Vi Utxo,... ,xi-ii, V~ = V2 Utyo,..., YI-iJ

md

E' = E U HU,Yi): u E Vi, i E (in U HU,Xi) : u E V2, i E (in. ::aearly Gi is dense. Now consider the set N = Mn+I(Gi) U Mn+l-i(Gi). 3y setting up explicit bijections, it is readily seen that

Theorem 3.9 (Broder) The problem of counting perfect matchings in

IMn+I(Gi)1 = (l!)2IMn_I(G)1 (3.3)

IMn+l-i(Gi)1

The reader may be wondering at this stage whether the problem of counting perfect matchings remains hard when restricted to dense graphs: if not, of course, the approximation results of this section would not be very exciting. The following result, which is proved in (16J, serves to justify our approach.

(i!)2(2lIMn_I(G)1 + IMn-I+1(G)1

dense bipartite graphs is #P-complete. 0 So far in this section we have concentrated exclusively on biparite

+ (l + 1)2IMn-l-i(G)1) .

graphs because of their connection with the permanent. The Markov

rhis suggests a procedure for estimating the ratio IMn-I(G)I/IMn-l-i(G)I:

iefine Pi = IMn-I(G)I/INI, P2 = IMn-l-i(G)I/INI. By simulating the \1arkov chain MCpm(Gi) repeatedly from the same initial state, genertte almost uniformly, within smal tolerance ~, some number of elements )f N, and let Si, S2 be the proportions of the sample which correspond to Ilatchings in G of size n - L and n - L - 1 respectively. Then the quantity U+ 1)2 si!(2l+ l)s2 is an estimator ofthe desired ratio Pi!P2. Provided this

lS sufciently accurate for each l, the product of the estimated ratios gives a

~ood approximation to IMn(G)I. More precisely, for any specified accuracy E E (0,1) we can arrange for Si, S2 to approximate Pi,P2 respectively within

ratio 1 + €/4n with probabilty at least 1 - I/Sn. Repeating this for each l, the final estimate approximates IMn(G)1 within ratio (1 + €/4n)2n ~ 1 + € with probabilty (1 - I/Sn)2n 2: 3/4. To see that the necessar accuracy can be achieved in polynomial time,

note first that

chain MCpm(G) can be applied without essential modifcation to arbitrary graphs G. In fact, the only point at which we have relied on the bipartite structure of G is in the definition of the sets lC(M) in the proof of Theo-

rem 3.5 and the bound on their size. Let G = (V, E) be an arbitrary graph with !VI = 2n. As before, we assume that G contains a perfect matching. Call G dense if its minimum vertex degree is at least n. This ensures that

lC(M) for ME Mn(G) is stil well defined, and that IlC(M)1 ~ 2n2. The rest of the proof caries through as before, yielding b = Sn4 and consequently ¡i(H) 2: 1/16n4. The conductance is therefore bounded below by 1/64n6. (This can again be improved if transitions are implemented by random vertex selection.) Since a construction analogous to that of Corollar 3.7 holds for general dense graphs, we have

Corollary 3.10 There exists a f.p. almost uniform generator and a J.P. randomised approximate counter for perfect matchings in arbitrary dense graphs. 0

!. -c IMk(G)1 -c n2 n2 - IMk+1(G)1 -

for 0 ~ k -c n.

(3.4)

The lower bound is trivial, while the upper bound follows from the density

assumption in the same manner as the bound on IlC(M)1 in the proof~f Theorem 3.5. Hence from (3.3) the proportion of matchings in N corre~ sponding to (n - l)-matchings in G is at least

~ ~-.

(i!)2(2l + 1)IMn-I(G)1 2l + 1 1

INI - n2((l + 1)2 + 1) + 2l + 1 - 3n3

We conclude this section by examining the rôle played in our results by the density assumption. In the reduction of Corollary 3.7, we used it to prove the polynomial upper bound (3.4) on the ratios IMk(G)I/IMk+1(G)I.

The proof of Theorem 3.5 makes use of an even stronger property of dense graphs, namely that Mk (G) can be paritioned into classes of polynomially

bounded size, one for each element of Mk+i(G), such that al matchings in a given class are "close to" the corresponding element of Mk+1(G). In fact, it wil turn out that everything works under the considerably weaker

3.2 APPROXIMATING THE PERMANENT

10

18sumption that

IMn-i(G)1 :: q(n)IMn(G)1

(3.5)

or some fied polynomial q, where 2n is the number of vertices in G. First we show that this condition implies a similar bound on al the ratios Mk

(G)lIMk+i

(G)

I, generalising (3.4). This follows from the surprising

md beautifu fact that the sequence IMk(G)1 is log-concave, as we now

lemonstrate. ~emma 3.11 For any graph G and positive integer k,

3.2 APPROXIMATING THE PERMANENT

Clearly, the number of elements of Br reachable from a given (M, M') E Ar is just the number of M-paths in M EB M', namely r + 1. Conversely, any

given element of Br is reachable from precisely r elements of Ar. Hence if IArl ). 0 we have IBrl

ound in ¡43, Theorem 7.11 (see alo ¡65, Exercise 8.5.10n. We present m elementary combinatorial proof which uses ideas seen elsewhere in this :hapter. Since log-concavity results in combinatorics tend to be rather hard

o come by, we believe the simpler proof to be of independent interest. Jet k E N+. We may assume that IMk+1(G)1 ). 0, since the inequality

s trivially true otherwise. Define the sets A = Mk+i(G) x Mk-i(G) and

r+l ). 1,

r which completes the proof of the lemma. D IArl

Remark: In ¡43J, the tight inequalty

IMk+i(G)IIMk-i(G)1 :: IMk(G)12.

:'roof: A proof which relies on machinery from complex analysis can be

81

IMk(G)12 2: (k + ~~~m -,~ + 1) IMk+1(G)IIMk-i(G)1

is proved, where m = r n/21 and n is the number of vertices in G. The bound in our proof can also be improved with a little more care, but we will not labour this point here as simple log-concavity is quite adequate for

our purposes. D Next we show that (3.5) is sufcient to ensure rapid mixng for the family of Markov chains MCpm(G).

1 = Mk(G) x Mk(G). Our aim is to show that IAI :: IBI.

~ote fist that, for any two matchings M, M' in G, the symmetric diference kf EB M' consists of a set of disjoint simple paths (possibly closed) in G.

Theorem 3.12 For any graph G = (V, E) with IVI = 2n and IMn(G)1 ). 0, the conductance of the underlying graph of MCpm (G) is bounded below by

Jet us cal such a path an M -path if it contains one more edge of M

1 (IMn(G)I)2 161EI IMn-i(G)1

han of M'; an M'-path is defined similarly. Clearly, al other paths in kf EB M' contain equal numbers of edges from M and M'. Now for any pai M, M') E A, the number of M-paths in M EB M' must exceed the number

,f M'-paths by precisely 2. We may therefore partition A into disjoint lasses tAr: 0 oe r :: kì, where

1r=t(M,M') E A: M EB M' contains r+l M-paths and r-l M'-pathsì. :imilarly, the sets t Br : 0 :: r :: k ì with

Br=t(M,M') E B: MEBM' contains r M-paths and r M'-pathsì

iarition B. The lemma wil follow from the fact that IArl :: IBrl for

Proof: The proof is similar to that of Theorem 3.5 with one or two additional technicalties. As before, let H be the graph defining the random wal performed by MCpm(G) and N its state space. We wil need a variant of the canonical path counting argument which deals only with certain types

of paths. For a subset 8 of states with 181 :: INI/2, let S = N \ 8 and define 8i = 8 n Mn(G);

8~ = S n Mn(G);

82 = 8 n Mn-i(G);

8~ = S n Mn-i(G).

,et us cal a pair (L, L') E Br reachable from (M, M') E Ar if L EB L' =

We claim that, when counting paths crossing the cut from 8 to S, it is enough to consider paths from 8i to 8~ and from 82 to 8~. To make this precise, write r = IMn(G)I/IMn-i(G)1 and Q = 18il/1821, and let

¡f EB M' and L is obtained from M by taking some M-path of M EB M' and

Pi = 18i118~1 and P2 = 182118~1 be the numbers of paths from 8i to 8~ and

jpping the parity of al its edges with respect to M and M'. (This is anal-

from 82 to 8~ respectively. Clearly, Pi and P2 are respectively increasing

gous to unwinding the path in the maner of the proof of Theorem 3.5.)

and decreasing functions of Q (for fied r, 181, IN!); moreover, it is easy

ach r ). O.

82

3.2 APPROXIMATING THE PERMANENT

to check that they are equal when Q = r. Hence maxtPi,P2) is minimised when Q = r, and the minimum value is

3.2 APPROXIMATING THE PERMANENT

83

The final result is obtained via Proposition 2.11, the transition probabilties in MCpm(G) being 1/2IEI. (As before, this can be improved using a more

intellgent implementation.) 0 rI81(INI-181) :; rl811NI

(1 + r)2 - 8 since clearly r S 1.

Now all we need do is define canonical paths in H between elements of Mn(G) (perfect matchings) and elements of Mn-i(G) (near-perfect matchings). If no more than blNI of these paths use any oriented edge of H, by analogy with (3.1) we wil get the bound

Corollary 3.13 There exists a J.P. almost uniform generator and a f.p. randomised approximate counter for perfect matchings in any family of graphs satisfying (3.5) for some fied polynomial q.

Proof: The generator is immediate from the foregoing theorem. The

1 ( IMn(G)1 )

jL(H) ;: 8b IMn-i(G)1 .

(3.6)

counter follows via the reduction of Corollar 3.7, once we have noted from (3.3) that the graphs G¡ inherit from G a bound of the form (3.5) (for a diferent polynomial q). 0

Henceforth we consider only paths from perfect to near-perfect matchings, the complementary case being symmetrical. As in the proof of Theorem 3.5, the canonical path from I E Mn(G) to F E Mn-i(G) is determined by the symmetric difference I EI F, which now consists of an ordered sequence Oi, . . . , Or of disjoint cycles together with a single open path 0 both of whose endpoints are matched in I but not in F. The canonical path from I

to F proceeds by unwinding first the Oi as before and then 0 in the obvious way, with one endpoint nominated as start vertex.

Now let t be a transition in MCpm(G) from M to M' and P(t) ç Mn(G) x

F) E P(t),

Mn-i(G) the set of canonical paths which contain t. For (I, ~e define the encoding at (I, F) exactly

as in the proof of Theorem 3.5. A

noments reflection showd convince the reader that at I\n-2(G), since we now get an additional pair of

(I, F) E Mn-i (G) u unmatched vertices arising

rom the open path O. (Note that this takes us outside the state space, mt Lemma 3.11 wil take care of this.) Recovery of I and F from t and rt(I, F) works essentialy as before. Hence at is an injective mapping and ve have, using Lemma 3.11,

IP(t)1 S IMn-i(G)1 + IMn-2(G)1

Remark: As observed by Dagu et al (22), the reduction from counting to generation described in Corollar 3.7 may be replaced by the following

mechanism, with a smal increase in effciency. In analogous fashion to MCpm(G), we may define for 1 S k S n a Markov chain MCk(G) whose states are k- and (k-l)-matchings in G. (Thus MCn(G) is

just MCpm(G).) By a straightforward extension of the proof of Theorem 3.12, whereby mw-

tiple rather than unque canonical paths between states are alowed in the manner of Example 3.4, it can be shown that each of the chains MCk(G) is rapidly mixng under the same condition (3.5) on G. This alows the ratios IMk(G)I/IMk-i(G)1 to be estimated diectly for each k in turn. We do not dwell on this point here as we wi present a more natural algorithm for the permanent in the next section. 0

Our earlier reswts for dense graphs are now seen to be a special case of Corollary 3.13.2 We might well want to ask whether other natural classes of graphs exist which satisfy (3.5) for a suitable polynomial q and for which membership of a given graph in the class can be tested easily. One observation which is often usefu in this connection is the following. Let G be bi-

..IMn-i IMn(G)1 ( )1G (1 + IMn-i(G)I) = IMn-i(G)IINI

parite with 2n vertices. An alternating path for a matching ME Mn-i(G) is a path whose edges lie alternately inside and outside M. Suppose that G

IMn(G)1 . ~hus we may take b = IMn-i(G)I/IMn(G)1 in (3.6) to get

2Note that the density bound quoted is tight in the sense that it is possible to construct, for any fied 8 )0 0, a sequence of (bipartite) graphs (Gn)

jL(H)-:; 8! (IMn-i(G)1 IMn(G)1 )2

with 2n vertices and minimum vertex degree;: n/(2 + 8) such that the ratio IMn-i(Gn)/IMn(Gn)1 is exponentially large.

84

3.2 APPROXIMATING THE PERMANENT

has maxmum vertex degree d, and that any M E Mn-i(G) can be extended to a perfect matching in G by augmentation along an alternating path oflength at most 2l+1. Then the ratio IMn-i(G)I/IMn(G)1 is clearly

bounded above by nd i. (Note that this is essentially the mechanism we used for dense graphs, with L = 1.) Thus G satisfies (3.5) for some fied polynomial q(n) if L = O(IOgd n). This observtion enables one to show

3.3 MONOMER-DIMER SYSTEMS

85

Of perhaps greater interest, however, is the fact that the condition (3.5) can be tested for an arbitrary graph in randomised polynomial time, as we shal see in the next section. Thus we are able not only to approximate

the permanent in almost all cases, but alo to reliably identify the diffcult instances. This property gives our algorithm a pleasing robustness that is often not exhbited by algorithms that work for random instances.

that alost all graphs in fact satisfy (3.5). The following strong statement

of this fact is due to Mark Jerrum.

This concludes our discussion of perfect matchings for now. An alternative view of these results wil emerge as a by-product of our work on a

Theorem 3.14 (Jerrum) Let G = (Vi, V2, E) be a random bipartite graph

dierent problem in the next section.

with IVi! = IV21 = n, where each edge is selected independently with proba-

bility p;: (360Iogn)/n. Then, with probability 1 - O(n-i), G satisfies !Mn-i(G)1 ~ n4+o(i)IMn(G)I. 0

The crux of the proof is that a.e. such graph has maxmum degree O(pn) and augmenting paths of length O(Iogpn n). Combining Theorem 3.14 and Corollary 3.13, we immediately have the following result.

3.3

Monomer-climer systems

This section is concerned with counting and generating all matchings (independent sets of edges) in a graph. Apart from their inherent interest, these problems arise in the theory of statistical physics, which is a rich source of combinatorial counting and generation problems.

A monomer-dimer system consists of a graph G = (V, E), which is

Corollary 3.15 There exists a f.p. almost uniform generator and a J.P.

usualy some form of reguar lattice, whose vertices represent physical sites;

rundomised approximate counter for perfect matchings in a. e. random bi7artite graph as in Theorem 3.14. 0

adjacent pairs of sites may be occupied by diatomic molecules, or dimers.

)f course, Corollar 3.15 implies the existence of a f.p. randomised ap-

Confguations of the system correspond to arrangements of dimers on the lattice in which no two dimers overlap. In a confguation consisting of k ~ IVII2 dimers, unoccupied sites are referred to as monomers, and the

)roximation scheme for the permanent of a.e. random 0-1 matrix in the

ratio (I V I - 2k) / I Vi is the monomer density. Monomer-dimer systems have

malogous modeL.

been extensively studied as models of physical systems involving diatomic molecules. In the two-dimensional case, the system models the adsorption

lemark: The bound on the density p in Theorem 3.14 is, up to a constant actor, equal to the threshold vaue for the existence of a perfect matching in uch graphs (14), and hence alost best possible. By a more carefu appli-

ation ofthe same ideas, Corollar 3.15 may be extended to random graphs

of dimers on the surface of a crystal. Three-dimensional systems occur

in the theory of mixtures of molecules of different sizes and in the cellcluster theory of the liquid state. For fuher information, see (43) and the references therein.

f arbitrary density. Specifcaly, if G = (Vi, V2, E) is a random bipartite raph with IVi! = IV21 = n and

any edge density p, then with probabilty

~nding to 1 as n tends to infnity, either G contains no perfect matching or 'in-i(G)1 ~ nioIMn(G)I. (For the details, see (50).) Since the existence f a perfect matching can be tested in polynomial time, we conclude that

:orollary 3.15 holds for random bipartite graphs of any density. 0

Other classes of graphs can often be tackled using the same approach. Dr example, Dagum et al (22) have shown the existence of augmenting

aths of constant length for al en-reguar biparite graphs for any fied )- O. Other simple deterministic criteria ensuring (3.5) doubtless exist.

Most thermodynamic properties of such a system can be deduced from knowledge of the number of possible confguations, which is just the num-

ber of matchings in G. More generaly, each edge e of G has an associated weight c(e) E lR+ which represents the relative probabilty of occupation by a dimer. This wil depend on the contribution of such a dimer to the global energy of the system. The quantity of interest is the partition function

Z(G) = L W(G,M), MEMATCH(G)

(3.7)

where MATCH(G) is the set ofmatchings in G and W(G, M) = TIeEM c(e) is the weight of M. Counting matchings, Le., the special case

of (3.7) in which

3.3 MONOMER-DIMER SYSTEMS I edge weights are unity, is a #P-complete problem even when restricted planar graphs (46, 97J. The main result of this section is that the more neral sum (3.7) can in fact be approximated effciently for any weighted aph G.

We shal again proceed via a related generation problem for matchings. tlce the sum in (3.7) is weighted, however, matchings should be generated

it uniformly but with probabilties proportional to their weights. In fact, mpling monomer-dimer confgurations from the weighted distribution is

. interesting problem in its own right as a means of estimating the expection of various physical operators. We begin by generalsing the concepts

Chapter 1 a little to handle weighted structures.

Let R be a relation over E, and W : E* x E* __ lR an arbitrary funcm. For any problem instance x E E* and solution y E R(x), we cal (x,y) the weight ofy. We shal always assume that the weights are sim~ in the sense that W is computable in polynomial time. The weighted

3.3 MONOMER-DIMER SYSTEMS

87

where e is any edge of G and G-, G+ are as in Example 1.1. In the case that al solution weights are positive, a corresponding

weighted generation problem for (R, W) may be defined: this requires that each solution y E R(x) be output with probabilty proportional to its weight W(x, y). The vaious definitions of generators given in Section 1.2 carry over in an obvious way to the non-uniform case. In particular, our main notion of tractabilty is an almost W-generator g for R, which is defined exactly as for an alost uniform generator except that, for all inputs (x, €)

and solutions y E R(x), the output probabilties must satisfy (1 + €)-lø(X, €)W(x, y) :: Pr(g(x, €) = y) :: (1 + €)Ø(x, €)W(x, y)

for some function ø : E* x lR+ -- (0, IJ. As before, g is fuy-polynomial iff its runtime is bounded by a polynomial in ixl and 19 c1.

m LYERCx) W(x,y) wil be denoted by #wR(x). The concept of self-

The results of Section 1.3 transfer directly to the weighted case: thus we can work freely in the OCM model when considering approximation

:lucibilty can be extended to the weighted case as follows. We cal the ir (R, W) self-reducible if R satisfies conditions (srl)-(sr3) of Section 1.1

algorithms. Fuhermore, it is easy to verify that, with minor modifications, the reductions of Theorems 1.10 and 1.14 also hold for weighted problems,

d in addition (with the same notation):

giving the following generalisation of Corollary 1.15.

~r4) There exists a polynomial time computable function g : E* x E* x lR -- lR such that, for al x E E*,

#wR(x) = L g(x,w,#wR('l(x,w))). wEEU("')

us definition merely generalises the idea that #R(x) can be computed

Corollary 3.16 Let R ç E* x E* and W : E* x E* -- lR+. If the pair (R, W) is self-reducible, then the following are equivalent: (i) There exists a J.P. almost W-generator for R.

(ii) There exists a f.p. randomised approximation scheme for #w R. o

,ily given the vaues of #R for a few smaller problem instances. The ighted counting problem for the pai (R, W) involves computing the fucn #w R. By analogy with the unweighted case, this problem wil be rerded as tractable if there exists a f.p. randomised approximation scheme . #w R, as defined in Section 1,2. The monomer-dimer partition function (3.7) can be expressed.as the ighted counting problem for a self-reducible pair as follows. Let MATCH

the relation which associates with a (weighted) graph G al matchings G; this relation was seen to be self-reducible in Example 1.1. Let c(e) tlote the weight of the edge e in G, and for M E MATCH(G) define'(G, M) = IleEE c(e) as in (3.7). Then the pair (MATCH, W) is self-

lucible since

#WMATCH(G) = #wMATCH(G-) + c(e) #wMATCH(G+),

Hence we wil get a f.p. randomised approximation scheme for the

monomer-dimer partition function provided we can generate matchings with probabilties roughly proportional to their weights. This we achieve

using a suitable Markov chain simulation. Given a graph G = (V,

E) with positive edge weights tc(e): e EEl, we

consider the Markov chain MCmd (G) with state space N = MATCH( G) and

transitions as follows: in any state MEN, choose an edge e = (u, v) E E uniformly at random and then (i) if e E M, move to M - e with probabilty 1/(1 + c(e)) (Type 1 transition) ;

(ii) if u, v are both unmatched in M, move to M +e with probabilty c( e) / (1 + c( e)) (Type 2 transition);

88

3.3 MONOMER-DIMER SYSTEMS

(üi) if e' = (u,w) EM for some w, and v is unmatched in M, move

3.3 MONOMER-DIMER SYSTEMS

89

that b here is precisely a weighted analogue of the bottleneck parameter

to (M + e) - e' with probabilty c(e)/(c(e) + c(e')) (Type 0

we have been using in the previous two sections. Taking (3.8) and (3.9) together, we have the following bound on the ergodic flow out of S, where cut(S) denotes the set of transitions crossing the cut from S to S:

transition) ; (iv) in al other cases, do nothing.

As always, we simplify matters by adding a self-loop probabilty of 1/2 to each state. It is then readily checked that MCmd(G) is irreducible and ape-

riodic, and hence ergodic. Moreover, by considering the detailed balance equations (trl) on page 45, it is easy to see that the chain is reversible and

Fs = L Wt 2: b-i L L 7r¡7rF tEcut(S) tEcut(S) (I,F)EP(t) 2: b-i L 7r¡7rF

¡ES

FES

that the stationar probabilty 7rM of M E MATCH(G) is proportional to

its weight W (G, M) = TIeEM c( e). Thus the simulation procedure of Figure 2.1 will yield a f.p. almost W-generator for MATCH provided MCmd(G)

is rapidly mixng. Applying the analysis of Chapter 2, the crucial fact is

)- Cs. - 2b

By definition, the conductance of H therefore satisfies

the following. 1

Theorem 3.17 For a graph G = (V, E) with positive edge weights the underlying graph of the Markov chain MCmd(G) is at least 1/(8IElc~ax), where Cmax = max-(l, maxeEE c(e)J.

-(c(e) : e E Eì, the conductance of

Proof: Let H be the underlying graph of MCmd(G). The first step is to establish a weighted version of the path counting argument which led to the bound (3.1). Suppose that between each ordered pair (1, F) of distinct

q,(H) 2: 2b'

(3.10)

Our aim is thus to define a set of paths that give a suitably smal vaue for the bottleneck parameter bin (3.9).

To do this we generalise the proof of Theorem 3.5. Suppose there is an underlying order on al simple paths in G and designate in each of them a start vertex, which must be an endpoint if the path is not a cycle but

is arbitrary otherwise. For distinct 1, FEN, we can write the symmetric

states we have a canonical path in H consisting only of edges of non-zero

diference 1 EB F as a sequence Qi,. . . , Qr of disjoint paths which respects

weight (corresponding to valid transitions in the chain). Futhermore, let us

the ordering. The canonical path from 1 to F involves unwinding each of

!lsociate a weight 7r¡7rF with the path from 1 to F. If S is any non-empty

the Qi in turn as follows. There are two cases to consider:

mbset of states with capacity Cs = ¿MES 7rM ~ 1/2, the aggregated weight of al paths crossing the cut from S to S = N \ S satisfies

L 7r¡7rF = CsC'S 2: Cs/2.

¡ES

(i) Qi is not a cycle. Let Qi consist of the sequence (vo, Vi,..., vi) of vertices, with Vo the start vertex. If (vo, vi) E F, perform (3.8)

a sequence of Type 0 transitions replacing (V2j+1,V2j+2) by

as

transition if L is odd. If on the other hand (vo, vi) E 1, begin with a Tye 1 transition removing (vo, vi) and proceed as before

(V2ji V2j+1) for j = 0,1,. ", and finish with a single Tye 2

FES \¡ ow let t be a transition from a state M to a state M' '" M, and

ISUal denote by P(t) the set of al ordered pairs (1, F) whose canonical

iath contains t. Suppose it is known that, for any such transition t, the Lggregated weight of paths containing t satisfies

L 7r¡7rF ~ bwt,

(3.9)

(I,F)EP(t)

vhere Wt = 7rMPMM' = 7rM'PM'M is the weight of the edge in H corre-

:ponding to t. (PMM' is the transition probabilty from M to M'.) Note

for the reduced path (vi, . . . , Vi), (ii) Q i is a cycle. Let Q i consist of the sequence (vo, Vi, . . . , V2l+ i) of vertices, where L 2: 1, Vo is the start vertex, and (V2j, V2j+1) E 1 for 0 ~ j ~ l, the remaining edges belonging to F. Then the

unwinding begins with a Type 1 transition to remove (vo, vi). We are left with an open path 0 with endpoints vo, Vi, one of which must be the star vertex of O. Suppose Vk, k E -(0,1ì,

is not the start vertex. Then we unwind 0 as in (i) above but

90

3.3 MONOMER-DIMER SYSTEMS treating Vk as the start vertex. This trick serves to distinguish

paths from cycles, as wil prove convenient shortly.

~ow let t be a transition from M to M' =l M. The next step is to define our injective mapping O't : P(t) -- N. As in the proof of Theorem 3.5, we set 7t(I, F) equal to 1 E9 F E9 (M U M'), and remove the edge ei,t of 1 adjacent

3.3 MONOMER-DIMER SYSTEMS from which (3.11) follows. .

(ii) t is a Type 2 transition. This is handled by a symmetrical argument to (i) above, with M replaced by M'.

(ii) t is a Type 0 transition. Suppose M' = (M+e)-e', and consider

~onsists of independent edges, and so is an element of N. The difference

the multiset O't(I, F) U M. This is equal to the multiset 1 U F except that the edge e, and possibly also the edge ei,t are absent from it. Assuming ei,t is absent, which happens precisely when

r E9 F can be recovered from O't(I, F) using the relation

the current path is a cycle, we have

GO the start vertex of the path currently being unwound if necessar: this is

,0 iff the path is a cycle and t is Type O. It is now easily seen that O't(I, F)

1 E9 F = current path is a cycle;

7ri7rF = c( ei,t)c( e )7rM7rO't(i,F)

O't(1,F)F) U M'), otherwise. f (O't(1, E9E9 (M (M U M')) + ei,t, if t is Type 0 and the Note that we can tell whether the current path is a cycle from the sense of iinwinding. Recovery of 1 and F themselves now follows as before from the

path ordering. Hence O't is injective.

Moreover, it should be clear that O't(1, F) is very nearly the complement )f M in the union of 1 and F viewed as a multiset, so that the product 7ri7rF is approximately equal to 7rM7rO't(I,F), giving us a handle on the bottleneck

parameter b in (3.9). We now make this precise.

91

= c(ei,i)c(e)

(Wi/PMM' )7rO't(I,F)

= 2IElc(ei,t)(c(e) + c(e'))wt7rO't(I,F),

which again satisfies (3.11). If ei,t is not absent, the arguent

is identical with the factor c(ei,t) omitted. This concludes the proof of the Claim and the theorem. D

Claim For any (1, F) E P(t), we have 7ri7rF $ 4IElc~axwt7rO't(I,F) . (3.11)

The Claim wil be proved in a moment. First note that it immediately yields the desired bound b in (3.9), since for any transition t we have

L 7ri7rF $ 4lElc~axwt L 7rO't(i,F) $ 4IElc~axwt,

(i,F)EP(t) (i,F)EP(t)

where the second inequalty follows from the fact that O't is injective. We may therefore take b = 4IElc~ax' which in the light of (3.10) gives the ~onductance bound stated in the theorem. (t remains only for us to prove the Claim. We distinguish three cases: (i) t is a Type 1 transition. Suppose M' = M - e. Then O't(1, F) = 1 E9 F E9 M, so, viewed as multisets, M U O't(1, F) and 1 U Fare equal. Hence we have

Corollary 3.18 There exists a f.p. almost W-generator for matchings in arbitrary weighted graphs, where W is the weighting function for matchings defined above, provided the edge weights are positive and presented in unary.

Proof: Define Cmin = min-(1, mineEE c(e)ì. Then the minimum stationary

state probabilty in MCmd(G) is at least c~in2-IElc;;~, where n = IVI.

The logarithm of this quantity is at least -p(lxl), where ixl is the size of the input description and p is a polynomiaL. Hence by Theorem 3.17

and Corollary 2.8 the Markov chain family is rapidly mixng. Simulation an OCM is a simple matter, and we may take the empty

of MCmd(G) by

matching as initial state. D In view of Corollary 3.16, we may now state the main result of this section.

7ri7rF = 7rM7rO't(i,F)

Corollary 3.19 There exists a f.p. randomised approximation scheme for = (Wi/PMM')7rO't(I,F)

= 21EI(1 + c(e))wt7rO't(I,F)'

the monomer-dimer partition function of arbitrary weighted graphs with edge weights presented in unary. D

3.3 MONOMER-DIMER SYSTEMS

12

lemark: The special case in which all edge weights are unity corresponds o unweighted counting of matchings. Corollaries 3.18 and 3.19 therefore mply the existence of a f.p. almost uniform generator and a f.p. randomised

~pproximate counter for all matchings in arbitrary graphs. D As we have already mentioned, the monomer-dimer chain provides some

urther insight into the results of the previous section. In particular, it rields alternative and arguably more natural algorithms for generating and

:ounting perfect matchings in familes of graphs satisfying (3.5), and hence a iew proof of Corollary 3.13. The key to these algorithms is the introduction

if carefuy chosen edge weights. Let :F be a family of graphs satisfying (3.5) for some fied polynomial q, md G = (V, E) be a member of:F with ¡VI = 2n and IMn(G)1 :; O. We

ntroduce the notation G(c) to stand for the weighted graph obtained by ~signing to each edge of G the weight c. To generate perfect matchings n G almost uniformly, we proceed as follows: for some suitable C :; 1, use he Markov chain MCmd(G(C)) as above to generate matchings in G(c) (or !quivaently in G) from the distribution determined by the edge weights,

md fail if the ouptut is not a perfect matching. For any vaue of c, the nduced distribution on perfect matchings is clearly uniform. Now suppose v-e choose c = 2q(n)j since all edge weights are polynomially bounded, jorollary 3.18 ensures that the generator is f.p. It remains only to check hat the failure probabilty is not too large.

Writing mk = IMk(G)I, the log-concavity of the mk (Lemma 3.11) imilies that

mk =n-l II mi :: q(nt-k mn i=k mi+l

or 0 :: k :: n. In the stationary distribution of MCmd(G(C)), the probaiilty of being at a perfect matching is therefore

3.3 MONOMER-DIMER SYSTEMS

93

the ratios mk+i!mk in turn in a sequence of n stages, for k = 0, . . . , n - i. Since mo = 1, an approximation to mn is obtained as the product of the estimated ratios. The main idea is to compute a sequence Ci,..., Cn-l of edge weights with the property that, in the stationary distribution of the Markov chain MCmd(G(Ck)), the probabilty of being at a k-matching is

quite large, alowing the ratio mk+i!mk to be estimated statistically. That such a sequence of weights exists is a consequence of the log-concavity of the mk. Idealy, we want the weight Ck to be the inverse ratio mk-i!mk; however, it wil suffce to substitute the estimate of this quantity obtained in the previous stage.

Figue 3.2 shows the approximate counter for perfect matchings, where G is the input graph and E :: 1 the relative error specified for the final approximation. The algorithm begins by testing whether G contains a perfect matching, for which purose any standard polynomial time algorithm may be used. We therefore assume in the following discussion that

IMn(G)1 :; O. In line (4), Q is the alost W-generator for matchings described earlier, i.e., the cal Q(G(c), . ) invokes a simulation of the Markov

chain MCmd(G(C)). Line (2) and the iterations of the for-loop correspond to the n stages of the computation mentioned above. Let Ck+1 be the value

of the weight parameter c at the end of the kth stage: we claim that, for each k, Ck+1 approximates mk/mk+1 within ratio 1 + E/2n with high probabilty, provided the vaue t in line (4) is suitably chosen. This wil II output in line (8) approximates mn within ratio imply that the product (1 + E/2nt :: 1 + E with high probabilty, as requied.

The claim is proved by induction on k. The base case is trivial since, from line (2), Ci = IEI-I = mo/mi. Now assume inductively that Ck approxiates mk-i!mk within ratio 1+E/2n. In the stationary distribution of the chain MCmd(G(Ck)), let Pi, 0:: i :: n, denote the probabilty of

being at

an i-matching. Then clearly mk/mk+1 = CkPk/Pk+1' Inspection of line (7)

:; 2n 1

reveal that Ck+1 = CkPk/Pk+1, where Pk,Pk+1 are estimates of Pk,Pk+1

L 2k 2

rin-i!mn, very large edge weights would be required to ensure that perfec~

and (k+ 1)-matchings in a sample produced by the generator. Since the tolerance is e = E/30n, by makng the sample size t large enough we can ensure that these estimates are within ratio 1 + 5e = 1 + E/6n with high probability. Then Ck+1 approximates mk/mk+l within ratio (1 + E/6n)2 :: 1 + E/2n the proof. Note that with high probabilty, completing the inductive step of

aatchings appear frequently enough. ..

the pathological cases of lines (3) and (6) can occur only in the unlely

n k mn(2q(n)t

L mk(2q(n))

k=O

-- :; k=O

~his confrms that the method works. Notice how the condition (3.5) again

,rises naturaly here: in the absence of a polynomial bound on the ratio

Now let us review the problem of counting perfect matchings. The

computed in lines (4) and (5) by observng the proportions of k-matchings

event that some estimate Ck is out of range.

aonomer-dimer chain suggests a more natural reduction to generation than

Finaly, we need to investigate the runtime of the procedure. With the

hat of Corollar 3.7. For a graph G = (V, E) as above, we wil estimate

exception of lines (4) and (5), this is evidently bounded by a polynomial

3.3 MONOMER-DIMER SYSTEMS

(1)

satisfying (3.5).

c := lEI-I; II:= lEI;

for k := 1 to n - 1 do begin

Recall from Theorem 3.14 ofthe previous section that almost all graphs

if c :; 2q(n) or c.c (2IEI)-1 then halt with output 0

(3)

else begin (4)

the form g(G(c),E/30n)

Pk := IY n Mk(G)I/t; PHl:= iy n MH1(G)I/t; if Pk = 0 or PHI = 0 then halt with output 0

perfect matching, behaves as follows:

and let Y be the set of outputs;

(8)

satisfy (3.5) for a fied polynomial q. However, it is not at all clear how to decide whether a given graph satisfies the bound. The above techniques suggest a simple randomised procedure for doing this. More precisely, let q be an arbitrar polynomial and suppose we wish to design an effcient algorithm A which, when presented with an arbitrary graph G containing a

make t cal of

(5) (6) (7)

95

in nand E-1. Hence the procedure of Figure 3.2 constitutes a f.p. randomised approximate counter for perfect matchings in familes of graphs

if IMn(G)1 = 0 then halt with output 0

else begin (2)

3.3 MONOMER-DIMER SYSTEMS

else begin c:= CPk/PHl; II:= II/c end end end; halt with output II end

(i) if mn-i/mn ~ q(n), A accepts with high probabilty; (ii) if mn-i/mn :; 6q(n), A rejects with high probabilty.

Figue 3.2: Approximate counter for perfect matchings

For intermediate vaues of the ratio, we do not care whether A accepts or rejects. (The value 6 here is used for ilustrative purposes only and can

n. Moreover, the bound on the edge weights in line (3) ensures that each ii to 9 is polynomialy bounded in n and c 1. We just have to check that t :ed only be similarly bounded in order to give good estimates Pk, PH 1 with gh probabilty. To see this, assume that Ck is a good estimate of mk/mkH

id note that, for i ;: k, Pk

k k-ii-III mj mkCk

Pi

miCki = Ck . k mjH .

be replaced by any fied constant greater than 1.) Of course, as usual A

can also incorporate an effcient test for the existence of a perfect matching in G. Such an algorithm is of signficant practical value, since we can be almost

certain that any graph accepted satisfies mn-i/mn ~ 6q(n), alowing us to count and generate perfect matchings in it. Moreover, all graphs with

(3.12) .

i=

'ý log-concavity of the mi, each ratio in the product is bounded be-

w by mk-i/mk. Also, Ckk-i approximates (mk/mk_i)i-k within ratio + E/2n)i-k ~ (1 + l/n)n ~ e. It follows from (3.12) that Pk/Pi ;: l/e

r i ;: k. Exactly the same bound holds for i .c k. Since ¿Pi = 1,

~ conclude that Pk ;: (en + 1) - 1 and so is bounded below by an inverse

ilynomial in n. Moreover, we have Pk+l = Ckmk+l :;! (mo) ( mn ) :; --

Pk mk - 2 mi mn-i - 2IElq(n)

:ing log-concavity and (3.5), so that PHI is similarly bounded below. Iiese lower bounds imply by Proposition 3.8 (generalsed to the weighted se) that the sample size t required to ensure that Pk, PkH approximate

, PHI within ratio 1 +E/6n with high probabilty is polynomialy bounded

mn-i/mn ~ q(n) wil almost always be accepted. Taken in conjunction with Theorem 3.14, and with q(n) = n5 (say), this implies that we can not only handle almost al graphs in some reasonable model, but also reliably identify the pathological cases which are hard to deal with. As we observed earlier, this lends much greater signficance to our results about random instances. To obtain an algorithm A with the specified behaviour, all we need do is generate some number t of matchings in G using the chain MCmd (G(2q( n))) with some smal fied tolerance E, and note the proportion s of these which are perfect. We accept iff s ;: 3/8. As we have already seen, in the case

mn-i/mn ~ q(n) the expected value of s wil be :; 1/2(1 + E)2, so takng t = Cln8-i, for a suitable constant C and any desired 8:; 0, ensures that

Pr(s .c 3/8) .c 8. Similarly, if mn-i/mn :; 6q(n) the expected value of s is less than (1 + E)22q(n)/8q(n) = (1 + E)2/4, and Pr(s ;: 3/8) can again be

made arbitrarily small as above.

T U l¡

96

3.4 CONCLUDING REMARKS

3.4

Concluding remarks

As well as presenting several positive results, this chapter has left open a number of interesting questions which we now briefly mention.

The first question which suggests itself concerns the approximabilty of the unestricted permanent. Though based on dierent Markov chains,

both approximation algorithms we have employed here give rise naturally to condition (3.5), that the ratio of the number of near-perfect to perfect matchings must be polynomialy bounded. On the other hand, there seems to be no a priori reason why counting perfect matchings in graphs which

violate this condition should be particularly hard. Perhaps a radicaly different approach can be used to handle graphs of this kind. In any case, the task of devising an effcient unversaly vad approximate counter for perfect matchings, or alternatively showing that the problem is hard to

~

3.4 CONCLUDING REMARKS

97

fi ~ ~ li

!

i: "'0 ~ ~

avaable. Some improvements have aleady been achieved and are reported in the Appendix.

Similar considerations apply to our methods for estimating the expectation of a random vaiable under the stationar distribution of a Markov chain. We have chosen to view the chain as a generator of independent

samples, partly to simplify the statistical concepts and partly because the random generation problems are of interest in their own right. In contrast, Aldous (2) considers estimates derived by observng a Markov chain continuously, and in fact formulates a defition of rapid mixng directly in terms of the vaiance of such an estimate. This approach may lead to increased effciency.

Another important question is whether the techniques of this chapter can be applied to other interesting combinatOiial structures. In many cases,

approximate in general, in the manner of Theorem 1.17, seems to be a

a reversible Markov chain on the structures with the desired stationar dis-

thorny issue which may requie other technques for its resolution.

tribution suggests itself naturaly. We give a few examples as an ilustration.

As mentioned earlier, it would be of interest to know whether other easily characterised familes of graphs, apar from dense graphs, satisfy

Example 3.20 For a connected graph G, consider the Markov chain

(3.5) for some suitable polynomial q. For example, this question is important in the case of certain reguar lattice graphs, in which physicists are

keen to count dimer coverings (monomer-dimer confguations with zero monomer density). Much effort has been expended on this problem, and an elegant exact solution obtaied for planar lattices (or indeed arbitrar

planar graphs) (58). The three-dimensional case, however, remais open. Some smal-scale experimentation suggests that the ratio of near-perfect to perfect matchings is polynomialy bounded for three-dimensional rectangular lattices, alowing the counting problem to be effciently approximated,

MC( G) whose states are the spanning trees of G, with transitions as follows:

given a spannng tree T, select an edge e of G unformly at random. If e belongs to T, do nothing; otherwse, add e to T to form a graph with a sinthe cycle. Then MC(G) is gle cycle, and remove a randomly chosen edge of symmetric, so its stationar distribution is unform over the spanning trees of G. (As noted in Chapter 1, effcient exact methods do exist for counting and generating spanng trees. However, the simplicity of this Markov chain, together with the fact that it has certai features in common with

a number of other natural chains, make it an interesting object of study in

but we have not been able to confrm this analyticaly. (Note that the

its own right.) 0

straightforward augmenting path approach is of no help for such graphs due to their large diameter.) Another quantity of interest is the number of monomer-dimer confguations with given monomer density, i.e., the num-

Example 3.21 Let n be a positive integer and g = (go, . . . , gn-i) a degree natural numbers gi with Lgi even. Let sequence on (n), i.e., a sequence of

ber of matchings of given cardinalty. Both algorithms we have presented for counting perfect matchings can obviously be adapted to computeIMk(G)/ for any given k, but we are left with a similar condition on G, namely that

vertex i has degree 9i, and consider the Markov chain MC(g) defined as

the ratio IMk-i(G)//IMk(G)/be polynomialy bounded. From a practical point of view, it would be interesting to know whether the conductance bounds we have derived can be signficantly improved.

We make no claim of optimalty here, prefering to concentrate on giving a clear exposition of the rapid mixng property. The practical utilty of our algorithms, however, is likely to depend on rather tighter bounds being

GRAPHS(g) denote the set of al simple graphs on vertex set (n) in which

follows. The states of MC(g) are the elements of GRAPHS(g), together with all simple graphs on (n) whose degree sequence g' satisfies g~ ~ gi for all i

and Li Igi - g~1 = 2. (I.e., the degree sequence g' is obtained from g by a smal perturbation.) To make a transition from state G, select an ordered pai i, j of vertices uniormly at random and

(i) if G E GRAPHS(g) and (i,j) is an edge of G, delete (i,j) from G;

(ü) if G tt GRAPHS(g), the degree of i in G is less than 9i, and (i,j) is

98

3.4 CONCLUDING REMARKS

not an edge of G, add (i,j) to G; if this causes the degree of j to exceed gj, select an edge (j, k) uniformly

at random and delete it.

3.4 CONCLUDING REMARKS

99

Since alost al natural chains of this kind are reversible, the character-

isation of Chapter 2 can in principle be applied to obtain rigorous bounds on the number of simulation steps required to achieve any specified level of

)nce again, the stationary distribution is uniform over GRAPHS(g). (We

accuracy. We conjecture that this is possible for some of these chains, and

Iiall attack the generation problem for the relation GRAPHS by a completely

that the path counting technique developed in this chapter is a promising

ifferent method in the next chapter.) 0

approach for obtaining upper bounds. Indeed, as reported in the Appendix, progress has recently been made in the analysis of a Markov chain associ-

lxample 3.22 Let n be a positive integer and w a partial order on the

ated with the Ising model (52J.

~t (nJ. Let LE be the relation which associates with the pair (n,w) the

Finaly, let us briefly mention the widely used stochastic heuristic for combinatorial optimisation known as simulated annealing (60J. The basic idea is that a Markov chain explores a space of configurations (feasible solutions), each of which has an associated cost or "energy". In the stationary distribution of the chain, low cost solutions have large weight so the chain tends to favour them asymptotically. By progressively reducing a "temperature" parameter, the weights are scaled so as to accentuate the depths of

~t of al

linear extensions of w. A Markov chain MC(n,w) with state

iace LE( n, w) and uniform stationary distribution can be defined as follows: lven a linear extension J. of w, select an element of (n) uniformly

at random

n.d interchange it with the next largest element in J., provided the resulting

rder is stil consistent with w. (If it is not consistent, do nothing.) 0

,xample 3.23 Let G = (V, E) be a connected undirected graph, and ONN(G) the set of al connected spanning subgraphs of G, i.e., con-

~cted graphs (V, E') where E' ç E. Consider the Markov chain MC(G) hose state space is CONN( G) and which makes random transitions from

the energy wells. (Thus the process is not in general time-homogeneous.)

Whie such a process is known to converge asymptotically under faily general conditions (see, e.g., (41, 671), very little is known about its rate of convergence, and hence its effciency, when applied to non-trivial problems.

, E CONN( G) as follows: select an edge e E E uniformly at random; if e

Since the question is related to that of rapid mixng for an appropriate

ies not belong to H, add e to H; if e belongs to H, remove e from H

sequence of Markov chains, the methods described here potentialy provide

:ovided the resulting graph remains connected. This chain is easily seen i have uniform stationary distribution. If it could be shown to be rapidly ixng, it would yield an effcient randomised approximate counter for con~cted spanning subgraphs. This #P-complete counting problem has apications in the theory of network reliabilty (81 J. 0

The first three chains above have in fact recently been analysed and iown to be rapidly mixng, using the techniques developed in the last two iapters plus some additional ideas. A brief discussion and references are ven in the Appendix. The question of rapid mixng for the chain of Exnple 3.23, or any other chain that gives usefu computational information iout the relation CONN, remains open. Similar (usually non-unform) Markov chains have been the subject of tensive experimental investigation in statistical physics: examples include, If-avoiding walks (11 J and Ising confgurations (12J on various lattices. '

rpicaly, non-rigorous ad hoc arguments are given to justify the number of nulation steps performed in order to approximate the desired distribution i the states. Conclusions drawn from such experiments therefore often mand from the reader a certain act of faith.

a means of analysis.

Actually, the results of Section 3.3 do enable us to analyse a restricted form of simulated annealng algorithm for the problem of finding a maxmum cardinalty matching in a graph G = (V, E). (Although polynomial.time deterministic algorithms exist for this problem, they are far from simple.) The algorithm uses the chain MCmd(G(C)) for a suitably chosen (fied) vaue of the edge weight c. (In the above terminology, c is inversely

related to temperature.) Clearly, for c 2: 1 larger matchings have greater weight in the stationary distribution, and this effect is magnfied as c is increased. However, as we can see from Theorem 3.17, we pay a price for increasing c in the. form of a slower approach to stationarity. The idea is therefore to choose a vaue of c that is both smal enough to ensure rapid convergence and large enough so that the stationary distribution is concentrated on matchings of nearly maxmum size. To make this precise, suppose we are aiming to find a matching whose size is within a factor 1 - € of the size of a largest matching in G, for some

constant € E (0,1). We claim that, if we take c = 2IEI(1-e)/e, then the probabilty in the stationary distribution of being at a matching of at least this size is bounded below by 1/2. Since the vaue of c is polynomial in the

100

3.4 CONCLUDING REMARKS

size of G, we conclude from Theorem 3.17 that a polynomial time-bounded

simulation of MCmd(G(c)) wil fid a matching of the desired size with probabilty at least (say) 1/4. This can be boosted to 1 - 8 by repeating

Chapter 4

¡he experiment O(lg 8-1) times.

To justify the above claim, let ko be the size of a largest matching in G, md define k = r(1 - f)ko 1- Writing as usual mi for IMi(G)I, the log:oncavity property of matchigs (Lemma 3.11), together with the fact that nko ~ 1, implies that

Indirect Applications

ko

mk-l = mko II mj~l ~ (mk-1) ko-k+l

j=k mJ mk

(3.13)

In this chapter, we return to the general framework of Chapter 1 and re-

3ut since j-matchings in G are subsets of E of size j, there is alo the crude

examine the notions of approximate counting and alost unform genera-

ipper bound mk-l :S IElk-1. Henc~ from (3.13) we conclude that

tion in the light of our work on Markov chains. Our main result is a dra-

mk 2

matic improvement of the reduction from generation to counting for self-

mk-l :S IEI(l-e)/e = ~. (3.14)

l fuher application of the log-concavity property now shows that

ridmk :S (C/2)k-i for 0 :S i :S k, so the aggregated weight of matchings of ize less than k is

k-l k-l L mici :S (L 2i-k) mkck -( mkck . i=O i=O 'his ensures that the probabilty of being at a matching of size k or more : at least 1/2, as claimed.

A result similar to the above but with a much more complex proof is ¡ven by Sasak and Hajek (82). These authors also prove an interesting

egative result to the effect that no simulated annealing algorithm in this r a fairly large related class will fid a matching of maxmum cardinalty l polynomial time with high probabilty. There is undoubtedly consider)le scope for fuher positive ard negative results of these kinds; another

teresting example can be found in (47).

reducible relations presented in Theorem 1.10, which alows much larger errors in the counter to be handled. The reduction is achieved by con-

structing an ergodic Markov chain based on the tree of derivations. As always, the crucial feature of the chain from our point of view is that it converges rapidly to its stationary distribution. The machinery developed in Chapter 2 wil enable us to establish this property painlessly. This result has two major consequences. Firstly, it gives us a notion of approximate counting which is robust with respect to polynomial time computation in the following sense: for a self-reducible relation R, if ran-

domised approximate counting within ratio 1 + O(nß) is possible in polynomial time for some real constant ß, however large, then there exists a f.p. randomised approximate counter for R. In other words, a very crude counter can always be effectively bootstrapped to achieve arbitrarily good asymptotic behaviour. Secondly, it suggests a new technique for generating combinatorial structures when some asymptotic information about their number is avaable (to within a constant factor, say). As a concrete example of this technique in action, we consider the prob-

lem of generating labelled graphs with given vertex degrees and a given excluded subgraph. Using a result of McKay ¡69) which provides analytic

counting estimates for these graphs, we show that it is possible to generate them in polynomial time from a distribution which is very close to uniform

provided only that the maxmum degree grows no faster than O(m1/4), where m is the number of edges. In spite of the fact. that the problem is

apparently not self-reducible under this restriction, our technique can stil be applied with a little extra work. This result represents a considerable

102

4.1 A ROBUST NOTION OF APPROXIMATE COUNTING

4.1 A ROBUST NOTION OF APPROXIMATE COUNTING

103

improvement on hitherto known methods ¡13, 100). It also implies the ex-

backtrack from any vertex to its parent, the transition probabilities to all

istence of polynomial time randomised algorithms for counting such graphs ~o within a factor of 1 + m-ß, for any real ß.

adjacent vertices being proportional to the edge weights: thus we get a

dynamic process in which upward and downward movements are equally likely (except at the root and the leaves). To eliminate periodicity, we also

4.1 A robust notion of approximate

add to each state a self-loop probabilty of 1/2. Viewing this process as a symmetric random wal with reflecting barriers on the levels of the tree, it is easy to see that it converges rapidly (essentially in time polynomial in the depth of the tree) to a stationary distribution which is uniform over

counting The main aim of this section is to show how to construct a f.p. almost iiniform generator for a self-reducible relation given only very approximate ~ounting information for the structures concerned.

levels and also uniform over leaves. Hence a short simulation of the chain generates solutions almost uniformly, and the probabilty of failure can be made small by repeated trials. Now suppose that we have available only an

lorn path from the root of the tree of derivations to a leaf (solution): at

approximate counter for the structures in question, so that the edge weights in the tree are no longer accurate. Then we have grounds for optimism that this procedure might stil work effciently: the hope is that, since each edge weight infuences transitions in both directions, the process wil actualy be

~ach stage, the next edge is selected with probabilty proportional to the

self-correcting.

Let us first briefly review the straightforward reduction from generation ;0 counting used to establish Theorem 1.10. This involves choosing a ran-

lUmber of solutions in the subtree rooted at its lower end. By appending

t correction process based on the a posteriori probabilty of the path, we iaw that this procedure can be made to work even if the counter is ranlomised and slightly inaccurate, specificaly if

it is within ratio 1+0(n-kR),

;vhere kR ): 0 is a constant depending on R. However, when only rather

:ruder counting information is available (to within a constant factor, say) ;he basic "one-pass" technique breaks down owing to the accumulation of irrors which are too large to be corrected.

One possible approach to coping with cruder counting estimates is to )revent the accumulation of large errors by applying a correction process at 'requent intervs as we move down the tree. The effect of such a modifica,ion is to introduce an element of backtracking into the algorithm, since the

:orrection process works by throwing away, with some appropriate probLbilty, some final segment of the path aleady chosen. This suggests a nore flexible dynamic approach, based on a Markov chain, which we now

We now proceed with the details of the above construction. Let R be a self-reducible relation over the alphabet L, and suppose that we are given

a polynomially time-bounded approximate counter C for R within ratio p(n) = 1 + O(nß) for some ß E JR. Thus the error ratio in C need not even be constant, but may increase polynomially with the problem size. Since R

is self-reducible, C can always be modified so as to give an exact answer

(which wil be either 0 or 1) when its input is an atom; also, its output may always be rounded up to the nearest integer at the cost of adding at

most 1 to p(n). We shall assume throughout that C incorporates these modifications. We may also assume without loss of generality that p is monotonically increasing. To begin with, we shal consider the case where C is deterministic; the additional technical problems posed by randomised counters wil be dealt with later. We adopt the terminology and notation for self-reducible relations in-

lescribe. The credit for proposing this approach goes to Mark Jerrum. As ve shall see, the results of Chapter 2 enable us to analyse it quite easily md show, perhaps surrisingly, that it is effcient.

troduced in Section 1.1. Let x E L* be a problem instance with R(x) f: ø. Our aim is to set up an ergodic Markov chain MC(x) whose states are the vertices of the tree of derivations TR(x) and whose stationary distribution is uniform over the leaves of the tree. Let V, E be the vertex and edge sets

Consider again the reduction of Theorem 1.10. If the counter is exact,

respectively of TR(x), and set m = lR(x), r = p(lxl). Note that both m

ve may view it as assigning to each edge of the tree of derivations a~ nteger weight equal to the number of leaves in the subtree rooted at its.

and r are polynomially bounded in lxi, and that the depth of the tree is at

most m. For each edge (u, v) E E, define the quantity

ower end; the process can then be seen as an absorbing Markov chain in

I'hich the transition probabilties from any vertex (state) to its children

a-e proportional to the corresponding edge weights. Suppose now that he process is no longer constrained to move downwards, but may also

f(u,v)=fC(~nst(u)), ifvist.heparentofu; (4.1)

1. C(inst(v)), otherwise.

(Recal that inst( . ) gives the problem instance associated with any vertex

4.1 A ROBUST NOTION OF APPROXIMATE COUNTING

104

4.1 A ROBUST NOTION OF APPROXIMATE COUNTING

105

in the tree.) Since C is deterministic, f : E -+ N is a well-defined function

Thrning our attention to condition (mc3), since non-leaf states corre-

on E. The crucial property to bear in mind is that for any edge e E E, f(e) approximates within ratio r the number of leaves in the maxmal subtree

spond to failure of the generator we need to check that the stationary process wil be found at a leaf with reasonably high probabilty. This is

below e.

confrmed by the following lemma.

Next we define for each vertex v E V a degree

d(v) = L f(u,v). u:(u,v)EE

Lemma 4.1 In the stationary distribution of MC(x), the probability of (4.2)

Note that d( v) 2: 1 for all v E V, and that d( v) = 1 if v is a leaf because C is exact for atoms. For each ordered pair v, u of vertices, the transition probabilty Pvu from v to u is then defined to be

being at a leaf is at least 1/2rm. Proof: Note that the degree sum D over the tree TR(x) may be written

D = L d(v) = 2 Lf(e).

vEV eEE

Now consider the collection of edges at some fied level of the tree. By (4.1)

Pvu = 1/2, if u = v;

0, otherwise.

(4.3)

i f(u,v)/2d(v), ~f (u,v) E E;

Thus there is a non-zero transition probabilty between two states iff they

the weight of each such edge approximates within ratio r the number of leaves in the maxmal subtree rooted at its lower end. Since these subtrees are disjoint, the aggregated weight of al edges at this level is at most r#R(x). Summing over al

are adjacent in the tree. The self-loop probabilty 1/2 ensures that the

levels of the tree yields the bound

D = 2 L f(e) :: 2rm#R(x).

chain is aperiodic. It is clearly also irreducible, and hence ergodic, and the detailed balance equations (trl) of page 45 show that it is reversible! with stationary distribution 7r' = (-irv)vEv that is proportional to the degrees,

Since 1rv = 1/ D for each leaf v, the stationary probabilty of being at a leaf

i.e,

is

= #R(x) )- ~

d(v) 1rv = D "Iv E V,

leaves v, we immediately see that 7r' is unform over them. Identifying leaves with solutions, we

where D = ¿vEV d(v). Since d(v) = 1 for al

can therefore construct an alost uniform generator for R by applying the

simulation paradigm of Figue 2.1 to the family of Markov chains MC(x). The generator wil be fully-polynomial provided we can verify conditions (mc1)-(mc4).

The fist two conditions are easy: a single call to the counter will tell us

whether R(x) = ø, and we can always start the simulation at the root of TR(X). Individual steps can be simulated by cal to the counter, which allow both the local structure of the tree and the transition probabilties (4.3)

~o be inferred, in similar fashion to the algorithm of Figue 1.2. Since the '¡ze of

problem instance labels

in the tree never exceeds lxi, al calls wil be

(4.4)

eEE

L 1rv leaves v

D - 2rm'

as required. 0 Recall that m and r are each polynomialy bounded in lxi, so the probabilty of being at a leaf can be boosted to 1/2 by repeating the entire

experiment only polynomialy many times. This verifes (mc3).

We now address the trickier question of rate of convergence. Specificaly,

we want the family of chains MC(x) to be rapidly mixng, as stipulated in condition (mc4). Let us see whether the characterisation of Chapter 2 helps here. We have seen that the chain MC(x) is reversible. Moreover, by (4.4) the minium stationary state probabilty 1r~?n satisfies

(x) _- 1 1 1r. min D --)-2rm#R(x).

polynomialy bounded.

Since solutions are strings of length m over the fied alphabet E, we have #R(x) :: ¡Eim. This implies a polynomial bound of the form (2.12) on Iln fact, it is easy to see that the reversibilty property holds for any tree

Ig1r~?n' By Corollary 2.8, rapid mixg for the family MC(x) wi therefore

orocess, Le., any ergodic Markov chain in which the edges of non-zero weight in ;he underlying graph form a tree.

follow from a suitable lower bound on the conductance of the underlying graphs.

)6

4.1 A ROBUST NOTION OF APPROXIMATE COUNTING

4.1 A ROBUST NOTION OF APPROXIMATE COUNTING

107

emma 4.2 Let G be the underlying graph of the Markov chain MC(x)

where E(8) is the set of edges in 8. Since Cs = EVEs d(v)j D, putting

~fined above. Then the conductance of G is bounded below by

(4.5) and (4.'6) together yields

12 ' ips = Fs -C ~ -4

ip(G) ~ (4r2m)-i .

S r m

roof: Note first that in G each edge (u, v) E E has weight WUV =

(u,v)j2D, while the loop at v has weight d(v)j2D and all other edges ive weight zero. In what follows, we wil identify subsets of V with the ibgraphs of TR(X) which they induce. If 8 ç V is a subtree (connected

ibgraph) of TR(x), we let root(8) denote the vertex of 8 at minimum ,stance from root(V), the root of TR(x), and L(8) the number of leaves

which completes the proof of the lemma. D Lemma 4.2 ensures that the Markov chains MC(x) are rapidly mixng, verifying condition (mc4). Hence the simulation paradigm yields a f.p. almost uniform generator for R. We are now in a position to state the first

.8.

major result of this chapter.

L order to bound the conductance of G, we claim that it sufces to consider

Theorem 4.3 Let R be a self-reducible relation over~. If there exists

)ws out of al subtrees 8 with root(8) =l root(V).2 (Informally, the process

a polynomially time-bounded (deterministic) approximate counter for R within ratio p(n) = 1 + O(nß) for some real constant ß, then there exists a

il converge fast because it is quite likely to emerge from any such subtree, 'avellng upwards, within a small number of steps.) To see this, note first iat ip( G) ~ min ips, where the minimisation is over all non-empty subsets ç V with root(V) ft 8. But we may write any such 8 as the union

) U . . . U T¡ of disjoint subtrees no pair of which is connected by an edge

i TR(X), and we have

f.p. almost uniform generator for R. D The following example is a very simple application of Theorem 4.3. We shal come to a more significant application later.

Example 4.4 Consider the relation DNFSAT mentioned in Chapter 1,

Æ. _ Fs Ei FTi:; . FTi . Æ. ':S = -C = '" C - m~n-C = m~n':Ti'

S wi Ti i Ti i

which associates with a Boolean formula in disjunctive normal form its

ence it is clear that ip( G) ~ min ips, where the minimisation is now over

proximate counter for DNFSAT. Let ø = Di V ... V D¡ be a formula over n

1 subtrees 8 of TR(X) with root(8) =l root(V) as claimed.

vaiables, where each Di is a conjunction of literals, and for 1 :: i :: llet Si be the number of assignments to the variables of ø which satisfy Di. Note that the Si are trivial to compute: if Di contains k distinct literals and no contradictions then Si = 2n-k. But it is immediate that

set of satisfying assignments. The following argument provides a crude ap-

lower bound on ips for such subtrees is readily obtained. We may assume ithout loss of generalty that 8 is maxmaL. Then the flow out of 8 is just

's = j(cut(8))j2D, where cut(8) is the cut edge connecting 8 to the rest

¡

'the tree. But since j(cut(8)) approximates the number of leaves L(8)

#DNFSAT( Ø) :: L Si :: l #DNFSAT( Ø) ,

ithin ratio r, the flow is bounded below by Fs :; L(8)

- 2rD'

i=i (4.5)

'n the other hand, summing edge weights in the subtree 8 as in the proof r Lemma 4.1, we easily derive the bound

L d(v) = j(cut(8)) + 2 L f(e) :: 2rmL(8), (4.6) vES eEE(S) 21n fact, it is true for any reversible chain that, when computing q,( G), it is

iffcient to consider q,s only for sets S that form connected subgraphs of the ¡iderlying graph G (with edges of weight zero removed).

so that E Si approximates the number of satisfying assignments of ø within

the polynomial ratio l. By Theorem 4.3, this gives us a f.p. alost uniform generator, and by Theorem 1.14 a f.p. randomised approximate counter

for DNFSAT. (As mentioned in Chapter 1, such algorithms have been constructed directly by Jerrum, Valiant and Vazirani ¡53J and Karp and Luby ¡56J respectively.) D If the approximate counter in Theorem 4.3 is randomised, so that it may occasionally produce arbitrarily bad results, a similar reduction stil goes through but at the cost of some tiresome technicalities. We summarise the proof in this case.

18

4.1 A ROBUST NOTION OF APPROXIMATE COUNTING

heorem 4.5 The result of Theorem 4.3 still holds even if the approximate

iunter for R is randomised.

4.1 A ROBUST NOTION OF APPROXIMATE COUNTING

109

(see Section 1.1). We let V, V denote the vertex sets of TR(x) and TR(x) respectively. Note that V \ V consists of a union of disjoint maxmal subtrees of TR(x). Some modifications to the transition probabilties are alo

roof (sketch): Let x be a problem instance for which R(x) l ø. As ~fore, assume p is monotonic and set m = lR(x), r = p(lxl). We begin

r considering the intermediate case where the counter C is randomised it always produces estimates which are within ratio r of their correct ilues. We again define a Markov chain MC(x) on the tree TR(x), whose ansition probabilties are now determined as follows. Suppose the pro-

necessar. At vertex v E V, we compute vaues c( u), u E U U .¡ v ì, and d( v)

as before, where now U is the set of chidren of v in TR(x). If d(v) = 0 then we make a transition to the parent of v (if it exists) with probabilty 1/4, and remai at v with probabilty 3/4. Otherwise, we test whether

Eu c(u) ~ 4r2d(v): if so, we remain at v; if not, we make a transition to a neighbouring vertex with probabilties as in (4.7). (Note that the self-loop

!ss is currently at vertex v, and let U be the set of children of v.For

probabilty in each state is at least 1/2.) Once again, the leaves of TR(x)

tch u E U U .¡ v ì, make a call C (inst( u)) to the counter and denote the

are treated as a special case.

:srut c(u); then make a fuher independent set of cal C(inst(u)) for the

This chain is clearly ergodic on some subset of V containing V, namely those states which communicate with the root. Henceforth we redefine V to include only such states. The chain is also stil reversible because it is a tree process. Let us fist observe that the new vertices in V \ V have negligible effect. All transitions from V to V \ V occur with at most tiny probabilty 6, so if started in V the process is unlely to leave V during

~me vertices u and denote their sum d(v). Finaly, make a transition to

i adjacent vertex u with probabilty

f c(u)/4r2d(v), if u is a child of V; 1. c(v)/4r2d(v), if u is the parent of v,

(4.7)

id remain at v otherwise. (Note that the factor 1/4r2 ensures that these ansitions are always well-defined, and that there is a self-loop probabilty

the random vaiable d( v) at the root vertex v wi take the vaue 0 with

: at least 1/2 in each state; we have used 1/4 rather than 1/2 for consis-

probabilty very close to 1, thus causing the chain to leave the subtree

~ncy with the second part of the proof.) Clearly, if C is deterministic this ~duces (except for a uniform factor of 1/2r2) to the original chain. In the

mdomised case, it is easy to see that the transition probabilty Pvu from v i u is actualy the expectation

E 4r2d(v) = 4r2 E(J(u,v)) E d(v) ,

(!(u,V)) 1 ( 1 )

here the random vaiable f (u, v) is defined as in (4.1) and is independent , d( v ). The stationar distribution 7r' therefore satisfies

7rv oc I/E(d(v)-1)

Vv E V,

lId the fact that C is exact for atoms implies that d( v) = 1 with probabily 1 for leaves v. The chain is clearly stil reversible, and the rest of the roof goes through essentialy as in the deterministic case, with d( v) and

the course of the simruation. Shorud it enter a subtree in V \ V, however,

rapidly. In fact, it is not hard to see that the stationary probabilty 7rv of

a vertex v E V \ V is at most O(6k), where k is the distance of v from V in T R (x). As a resrut, the total weight of V \ V in the stationary distribution

is smal. Futhermore, the large exit probabilty from subtrees S ç V \ V ensures a lower bound on cI s similar to that in the proof of Lemma 4.2.

Examination of the transition probabilties within V reveal that we can view this portion of the chain as a chain of the restricted kind described in whose transition probabilties have been perturbed by a factor in the range (1 :l 6'), where 6' depends on 6 and can be made the first part ofthe proof

exponentialy smal in Ixl. It is then easy to see that the stationary probabilties of states in V undergo similarly smal perturbations in the range (1 :l 6,)m. As a resrut, a lower bound as in the proof of Lemma 4.2 also holds for subtrees S with root(S) E V, and so for al subtrees, which again implies that the conductance cI(G) is suitably bounded below. Assuming that the simruation starts at the root, we therefore get rapid convergence

(u, v) replaced by 1 /E (d( v) -1) and E (J (u, v)) respectively.

over the subset V of the state space,3 which is sufcient since V includes al

"ow suppose that the counter may in addition produce arbitrarily bad re-'

ilts with some smal probabilty 6: by Lemma 1.2 we may assume that ~ 2-p(lxl) for al problem instances in the tree, where p is any desired olynomial. Since we are no longer able to infer the structure of TR(x) "ith certainty, we must now work in the larger self-reducibilty tree TR(x)

3More precisely, we are using the r.p.d. 'óv(t) over V here, as defined in

Chapter 2. Note that Theorem 2.5 implies a suffcient condition for rapid mixng with respect to this measure also.

10

4.1 A ROBUST NOTION OF APPROXIMATE COUNTING

eaves of TR(x). A test applied to leaf labels ensures that no non-solutions

lIe output. D

4.1 A ROBUST NOTION OF APPROXIMATE COUNTING

111

to a simple criterion of approximabilty. We conjecture that this notion wil prove usefu in the future classification of counting problems.

It is natural to ask whether a similar state of afairs holds in the case of generation problems. Unfortunately, we know of no analogue of Thettemark: There is actualy a simpler way to prove Theorem 4.5, though ;he resulting algorithm is rather less natural and the process is

no longer

¡trictly a Markov chain. Note that the simulation of Theorem 4.3 can stil

ie performed using a randomised counter if we arrange to remember the

orem 4.6 for improving almost uniform generators with very large (poly-

nomial) bias. Nevertheless, Theorem 4.5 can be used to obtain a weaker bootstrapping theorem for generators.

mtputs of the counter on al previous cal so that each edge weight is

Theorem 4.8 Let R ç E x E* be self-reducible and kR be a constant as

:omputed at most once. Provided al values returned by the counter are :em 4.3 and our earlier analysis applies. By powering the counter, we can

inSection 1.1 such that, for all pairs (x,y) E R, Iyl = O(lxlkR). If there exists a polynomially time-bounded almost uniform generator for R within tolerance O(n-kR) then there exists a f.p. almost uniform generator for R.

msure that this condition fails to hold with vashingly smal probabilty, ;0 the effect on the overal process will be negligible. D

Proof: We know from Theorem 1.16 that a generator satisfying the above

tccurate within the given ratio, we are effectively in the situation of Theo-

hypothesis can be used to construct a polynomialy time- bounded ran-

A direct consequence of Theorem 4.5 is a powerful bootstrapping theo-

domised approximate counter for R within ratio O(nß) for some constant ß.

:em for approximate counters. Recall from Theorem 1.14 that a f.p. alost

An application of Theorem 4.5 now yields the result. D

iniform generator for a self-reducible relation R can be used to construct t f.p. randomised approximate counter for R. Thus, starting from an apJroximate counter within ratio 1 + O(nß), and proceeding via the reduc¡ion described in this section, we arive at a counter with arbitrarily good

On reflection, Theorem 4.8 is quite remarkable: it says that, by observing only polynomialy many outputs of a black box generator whose bias

=iymptotic behaviour.

bias. Indeed, the reduction is huge, from O(n-kR) to the exponential factor

is a fied function of the input size, we are effectively able to reduce this

exp( -nß) for any desired real ß. This seems counter-intuitive since such

rheorem 4.6 Let R be a self-reducible relation over E. If there exists a polynomially time-bounded randomised approximate counter for R within ratio 1+0(nß) for

some real constantß, then there exists af.p. randomised

approximate counter for R. D

an experiment can apparently yield only very local information about the output distribution. The non-trivial reduction of Theorem 4.5 seemingly plays a crucial rôle here. We close this section with an updated version of Corollar 1. 15 which

The chief signcance of Theorem 4.6 is that it establishes a notion Jf approximate counting which is robust with respect to polynomial time computation, at least for the large class of self-reducible relations. This is evident from the following.

Corollary 4.7 Let R ç. E* x E* be self-reducible. Then there .exists a

summarises what we know about approximate counting and alost unform

generation for self-reducible relations. Corollary 4.9 For a se1f-reducible relation Rover E, the following are

equivalent: (i) There exists a f.p. almost uniform generator for R.

polynomially time-bounded randomised approximate counter for R within

(ii) There exists a J.P. randomised approximate counter for R.

ratio 1 + O(nß) either for all ß E IR or for no ß E R D ',-.

We are therefore justified in calng the counting problem for a self-

reducible relation R tractable iff there is a polynomial time randomised procedure which with high probabilty estimates #R( x) to within a factor of the form 1 + O(lxIß), for some fied real ß. This provides a particularly convenient means of classifying #P-complete counting problems according

(ii) There exists a polynomially time-bounded randomised approximate

counter for R within ratio 1 + O(nß) for some real constant ß. (iv) There exists a polynomially time-bounded almost uniform generator

for R within tolerance O(n-kR), where kR is a constant as above. D

112

4.2 SELF-EMBEDDABLE RELATIONS

4.2 Self-embeddable relations

113

self-reducible relations, including DNF -satisfiabilty and natural restricted

This short section is in the nature of a caveat to the bootstrapping results we have just derived. It turns out that similar results hold for a rather trivial reason if the relation R in question has a property which we might term self-embeddability. Informally, R is self-embeddable if there exists a polynomial time computable function ~ which takes a pair Xl, X2 of

4.2 SELF-EMBEDDABLE RELATIONS

problem

instances and "embeds" them in an instance ~(Xi' X2), whose size is at most linear in IXil and IX21 and whose solution set is in (I-I)-correspondence with

the product set R(xi) x R(X2)' We also demand that, from each solution for ~(Xi' X2), the corresponding pair of solutions for Xl and X2 can be recovered easily. The relation MATCH of Example 1.1 is clearly self-embeddable: for

any pair Gi, G2 of graphs we may take ~(Gi, G2) to be the disjoint union of Gi and G2. A slightly less trivial example is the following.

Example 4.10 Consider the relation which associates with a directed graph G its set of (directed) Hamiltonian paths. Then given a pair Gi, G2

of graphs, the required embedding function ~ adds to their disjoint union a new vertex v, together with edges from v to al vertices of Gi and from al

versions of familar relations, such as Hamiltonian paths in planar graphs. Moreover, the Markov chain reduction technque presented earlier is quite natural and general and can sometimes be applied even in the absence of self-reducibilty. Evidence for this is provided by the relation GRAPHS dis-

cussed in the next section, which is apparently neither self-embeddable nor self-reducible under the degree restrictions imposed there.

How does self-embeddabilty impact upon almost uniform generation? Currently, the best answer we can give to this question is that, in the presence of both self-reducibilty and self-embeddabilty, a very strong bootstrapping mechansm for alost uniform generators is avaable: this is the subject of our next theorem. However, we know of no analogous result

when the self-reducibilty assumption is dropped. Theorem 4.12 Let R ç E* xE* be both self-reducible and self-embeddable. If there exists a polynomially time-bounded almost uniform generator for R within tolerance O(nß) for some real constant ß, then there exists a f.p. almost uniform generator for R.

vertices of G2 to v. 0

For self-embeddable relations, there is a simple bootstrapping mechanism for approximate counters. (This observation is due to Keith Edwards. It is also implicit in the work of several other authors, such as

Stockmeyer (91J.)

Theorem 4.11 Let R ç E* x E* be self-embeddable. If there exists a polynomially time-bounded (randomised) approximate counter for R within

ratio 1 + O(nß) for some real constant ß, then there exists a f.p. (randomised) approximate counter for R.

Proof: Let C be a counter for R as above within ratio 1 + p'(n), where p'(n) = cnß and c, ß )- O. Given a problem instance X E E* and a positive real �