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Probabilistic Combinatorics and Its Applications

AMS SHORT COURSE LECTURE NOTES Introductory Survey Lectures

published as a subseries of Proceedings of Symposia in Applied Mathematics

Proceedings of Symposia in

APPLIED MATHEMATICS

Volume 44

Probabilistic Combinatorics and Its Applications

Bela Bollobas, Editor

Fan R. K. Chung Persi Diaconis Martin Dyer and Alan Frieze Imre Leader Umesh Vazirani

xg American Mathematical Society v Providence, Rhode Island

LECTURE NOTES PREPARED FOR THE AMERICAN MATHEMATICAL SOCIETY SHORT COURSE

PROBABILISTIC COMBINATORICS AND ITS APPLICATIONS HELD IN SAN FRANCISCO, CALIFORNIA

JANUARY 14-15, 1991

The AMS Short Course Series is sponsored by the Society's Committee on Employment and Educational Policy (CEEP). The series is under the

direction of the Short Course Advisory Subcommittee of CEEP.

Library of Congress Cataloging-in-Publication Data Probabilistic combinatorics and its applications/Bela Bollobas, editor; [with contributions by] Fan R. K. Chung ... [et al.].

p. cm. — (Proceedings of symposia in applied mathematics; v. 44. AMS Short Course lecture notes.)

Includes bibliographical references and index. ISBN 0-8218-5500-X (alk. paper) 1. Combinatorial probabilities. 2. Random graphs. I. Bollobas, Bela. II. Chung, Fan

R. K., et al. III. Series. QA273.45.P76 1992 91-33123 519.2—dc20 CIP

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Table of Contents

Preface xiii

Random Graphs BELA BOLLOBAS 1

Constructing Random-Like Graphs F A N R . K . C H U N G 21

Discrete Isoperimetric Inequalities IMRE LEADER 57

Random Graphs Revisited BELA BOLLOBAS 81

Rapidly Mixing Markov Chains UMESH VAZIRANI 99

Computing the Volume of Convex Bodies: A Case Where Randomness Provably Helps MARTIN DYER AND ALAN FRIEZE 123

Finite Fourier Methods: Access to Tools PERSI DIACONIS 171

Index 195

Preface It has been known for many decades that in order to show the existence of

"peculiar" mathematical structures we need not construct them. Thus a sixty-years old result of Paley and Zygmund states that, if a sequence (cn)o° of reals is such that X^Lo cn = °° t n e n S^Lo ^c™ c o s nx ^a^s t o D e a Fourier-Lebesgue series for almost all choices of signs. Nevertheless, even now, sixty years later, no algorithm is known which, given any sequence (cn)o° with X^Lo cn = °°> constructs a single sequence of signs for which 5Z^=o ^cn c o s nx 1S n ° t a Fourier-Lebesgue series.

Another well-known example is that of a normal number: we do not know of any concrete normal number, i.e. a real number x which is such that for all natural numbers k and n > 2, in the base n expansion of x all possible blocks of k digits occur with approximately the same frequency. And this is in spite of the fact that it is known that almost every real number is normal.

Results of this kind are surprising but often not very deep: the second ex­ample is within easy reach of any undergraduate familiar with the rudiments of measure theory. What is considerably more surprising is that similar phenomena can be found in combinatorics, where we study simple, down-to-earth mathemat­ical objects, like graphs and hypergraphs. In fact, it is precisely in combinatorics that the 'probabilistic method' produces the most striking examples. Thus Paul Erdos, the main founder of probabilistic combinatorics, proved over thirty years ago that if log2 (™) < Q) — 1 then the Ramsey number R(s) = R(s, s) is at least n + 1. To see this, all we have to notice is that if we take all graphs on {1, 2 , . . . , n} then, on average, they have less than 1/2 complete subgraphs on s vertices, and so some graph on {1, 2 , . . . , n} has neither a complete subgraph on n vertices, nor a set of s independent vertices. To find explicitly such a graph is a very different matter.

Mostly due to the efforts of Erdos, probabilistic methods have become a vital part of the arsenal of every combinatorialist. Together with Renyi, Erdos also initiated the study of random combinatorial objects, mostly graphs, for their own sake, and thereby founded the theory of random graphs, which is still the prime area for the use of probabilistic methods. Over the years, probabilistic methods have become of paramount importance in many nearby areas, like the design and analysis of computer algorithms.

In its simplest form, as in the Erdos-Ramsey example above, the probabilis­tic method involves the use of the expectation of a random variable X on some probability space and relies on the trivial fact that if E(X) < m then at some

xii i

XIV PREFACE

point of the space, X takes a value less than m. In a slightly more sophisticated application of the probabilistic method, we make use of the variance and higher moments, together with Chebyshev's inequality and sieve methods.

The use of the expectation and higher moments remained the staple diet in probabilistic combinatorics for over two decades, but in recent years proba­bilistic combinatorics has undergone some revolutionary development. This is due to the appearance of exciting new techniques, such as martingale inequal­ities, discrete isoperimetric inequalities, Fourier analysis on groups, eigenvalue techniques, branching processes and rapidly mixing Markov chains.

The aim of the volume is to review briefly the classical results in the theory of random graphs and to present several of the important recent developments in probabilistic combinatorics, together with some applications. All the papers are in final form.

The first paper contains a brief introduction to the theory of random graphs. The basic models of random graphs are introduced and many of the fundamental theorems are presented. The proofs rely mostly on the expectation, variance and Chebyshev's inequality and, at the next level, on higher moments and sieve inequalities.

Many results from the theory of random graphs have found their way into computer science: random graphs are particularly useful in the design of algo­rithms. Although it is comforting to know that there are networks with all the required properties, it is considerably better to find explicit constructions for these networks. Thus there is a clear need for explicit constructions of graphs sharing many of the basic properties of various random graphs. The program of explicitly constructing random-like graphs is reviewed in the second paper. Graphs having a variety of useful properties are discussed (Ramsey, discrepancy, expansion, eigenvalue, etc.) and several explicit constructions are described (due to Paley, Margulis, Lubotzky, Phillips and Sarnak).

One of the most important recent developments in probabilistic combina­torics is the use of martingale techniques and discrete isoperimetric inequalities, and the exploitation of various 'concentration of measure' phenomena. Every space of random graphs of order n is naturally identified with a measure on the discrete cube with 2n vertices, so graph properties are identified with subsets of the cube.

Given a subset A of the cube, the ^-boundary A^ of A is the set of points within distance t of A. In an isoperimetric inequality on the cube, we wish to minimize the measure of ^4(*), keeping the measure of A fixed. If A^ is known to be large then the set (i.e. property) A is likely to be close (within distance t) of a random point of the cube (random graph). This indicates why discrete isoperimetric inequalities, to be discussed in the third paper, are of paramount importance in probabilistic combinatorics.

The powerful discrete isoperimetric inequalities and concentration of mea­sure type results often give much better results than the traditional expectation and variance method. In particular, there are many instances when one can prove that the probability of failure is exponentially small, while the standard methods

PREFACE xv

would give only polynomial bounds. One of the notorious problems that yielded to an attack along these lines is the chromatic number of random graphs. This is presented in the fourth paper, together with a beautiful inequality of Janson and the very important and powerful Stein-Chen method on Poisson approximation.

There are many natural probability spaces of combinatorial structures where we run into difficulties even before we can start. For example, if we wish to study random r-regular graphs of order n then the very first question we ought to answer is: about how many of them are there? If both r and n — r are large, say about n/2, then this is a rather difficult question. It would be satisfactory to generate our objects 'almost' uniformly (or according to whatever probability measure we wish to take) provided this generation is rapid enough to enable us to estimate the probability that the final random object (r-regular graph in the example above) has the property we are interested in.

Jerrum, Valiant and Vazirani proved in 1986 that approximate counting and approximate uniform generation are intimately connected. Furthermore, these questions are closely related to the 'mixing time' of a Markov chain associated with our problem. If this Markov chain is rapidly mixing, i.e. if it gets close to its stationary distribution in a short space of time, then efficient generation is possible. The aim of the fifth paper is to present a number of powerful new methods for proving that a Markov chain is rapidly mixing and to survey various related questions.

The next paper is also about rapidly mixing Markov chains and uniform generation, but the context is rather different. Given a convex body K in Rn, containing a small Euclidean ball and contained in a large Euclidean ball, in the presence of various 'oracles', how fast an algorithm can one give to approximate the volume of Kl In 1989 Dyer, Frieze and Kannan proved that there is a fast randomized approximation algorithm for approximating the volume; in fact, such an algorithm is provably faster than any deterministic algorithm. In addition to a full proof of an improvement on the previous results, Dyer and Frieze a number of applications of the algorithm, namely to integration, counting linear extensions and mathematical programming.

One of the most important Markov chains in combinatorics is the random walk on the cube. The convergence to the stable (uniform) distribution is best analysed with the aid of Fourier analysis, as shown by Diaconis, Graham and Morrison in 1989. The final paper starts with the basis of Fourier analysis relevant to the study of problems of this kind, and proceeds to several more sophisticated applications.

Throughout the papers, several unsolved problems invite the reader to do research in probabilistic combinatorics.

Bela Bollobas

Index

Affine Group, 186 almost every, 5 almost no, 6 almost surely, 5 alternating inequalities, 9 average degree, 18 ball, 73 basis, 118 Bernoulli random variable, 7 binary order, 77 binomial distribution, 7 boundary, 57 Canonical Path, 113 character, 179 chromatic number, 14 class function, 179 clique number, 13 colouring, 14 complementary point, 114 concentrated Levy family, 63 conductance, 27, 102, 103, 112, 131 congestion, 113 Constructing random-like graphs, 22 convex body, 124, 125 coset graphs, 36 Counting linear extensions, 162 cube, 171 cutoff phenomenon, 173 degree, 18 descent, 180 dimension, 176 discrepancy property, 22 discrepancy, 25 Discrete Fourier Transform, 171 down-set, 72 e- approximation, 127 edge magnification, 112 edge-boundary, 76 edge-isoperimetric inequality, 76 Ehrenfest urn problem, 175 eigenvalue property, 22 expander graphs, 26 expansion property, 22 expansion, 27, 108 explicit constructions, 35 extremal properties, 22

first-order sentences, 10 Fourier analysis, 174 Fourier inversions, 177 Fourier transform, 174, 176 fractional Hamming ball, 73 fractional set system, 73 fully polynomial randomized approxima­

tion scheme, 100, 132 generalized Paley sum graphs, 36 Generating uniform points, 153 G L n , 183 graph process, 5 grid graph, 70 group representations, 171 Hamming balls, 58 hitting times, 6 hook, 182

z-compressed, 61, 75 i-compression, 60, 74 i-sections, 60, 74 independence number, 13 induced, 25 Integration, 157 inversion theorem, 174 irreducible, 177 isoperimetric inequality, 58, 131 isoperimetric number, 77 Learning, 165 Levy family, 63 Levy mean, 64 linear extension, 130 log-concave function, 140 lower threshold function, 6 major access network, 41 Major index, 180 Margulis graphs, 37 Markov's inequality, 7 martingale 66 matroid, 118 Metropolis Algorithm, 190 monotone decreasing, 3, 73 monotone increasing, 3 neighbourhood of strong dependence, 94 non-blocking network, 41 normal number, 1 nth moment, 7

195

196

oracle model, 125 Orthogonal Group, 186 #P-ha rd , 129 Paley graph, 24, 35 Paley sum graphs, 36 Plancherel theorem, 174, 177 Poisson distribution, 8 Poisson measure, 93 Polytope Conjecture, 118 product graph, 70 property of graphs, 3 property, 4 quasi-random classes, 23 Quasi-concave functions, 160 Ramanunjan graphs, 39 Ramsey graphs, 24 Ramsey number, 12 Ramsey property, 22 Ramsey's theorem, 12 random graph processes, 5 random graphs, 2, 3 random subset, 88 random subsets, 4 random walk on graphs, 171 random walks, 133 rapidly mixing Markov chains, 131 representation, 176 r th factorial moment, 9

INDEX

set, 61 simplicial order, 59 standard deviation, 7 Stein-Chen Method, 93 stochastic programming, 164 strictly balanced, 18 strong membership oracle, 125 superconcentrator, 42 system, 75 t-boundary, 57 3-cycles, 192 threshold function, 6 total variation distance, 93 total variation, 172 transvections, 183 Upper Bound Lemma, 178 upper threshold function, 6 variance, 7 vertex colouring, 14 volume, 124 weak membership oracle, 126 weak separation oracle, 126 weight, 73 weighted cube, 72 Young tableau, 180 Young's semi-normal form, 189 Z§, 172