We prove two results concerning approximate counting of independent sets in graphs with constant maximum degree \Delta. The first implies that the Monte Carlo Markov chain technique is likely to fail if \Delta 6. The second shows that no fully polynomial randomized approximation scheme can exist if \Delta 25, unless RP = NP. 1 Introduction Counting independent sets in graphs is one of several combinatorial counting problems which have received recent attention. The problem is known to be #P-complete, even for low degree graphs [3]. On the other hand, it has been shown that, for graphs of maximum degree \Delta = 4, randomized approximate counting is possible [7, 3]. This success has been achieved using the Monte Carlo Markov chain method to construct a fully polynomial randomized approximation scheme (fpras). This has led to a natural question as to how far this success might extend. Here we consider in more detail this question of counting independent sets in graphs with constant m...

We define a new Markov chain on (proper) k-colourings of graphs, and relate its convergence properties to the maximum degree \Delta of the graph. The chain is shown to have bounds on convergence time appreciably better than those for the wellknown Jerrum/Salas--Sokal chain in most circumstances. For the case k = 2\Delta, we provide a dramatic decrease in running time. We also show improvements whenever the graph is regular, or fewer than 3\Delta colours are used. The results are established using the method of path coupling. We indicate that our analysis is tight by showing that the couplings used are optimal in a sense which we define. 1 Introduction Markov chains on the set of proper colourings of graphs have been studied in computer science [9] and statistical physics [13]. In both applications, the rapidity of convergence of the chain is the main focus of interest, though for somewhat different reasons. The papers [9, 13] introduced a simple Markov chain, which we shall refer to a...

This work is a continuation of [4]. The focus is on the problem of sampling independent sets of a graph with maximum degree ffi. The weight of each independent set is expressed in terms of a fixed positive parameter 2 ffi\Gamma2 , where the weight of an indepednent set oe is joej . The Glauber dynamics is a simple Markov chain Monte Carlo method for sampling from this distribution. In [4], we showed fast convergence of this dynamics for triangle-free graphs. This paper proves fast convergence for arbitrary graphs. Computer Science Division, University of California at Berkeley, and International Computer Science Institute. Supported in part by National Science Foundation Fellowship. 1 Introduction For a more general introduction and a discussion of related work we refer the reader to the companion work [4]. The aim of this work is given a graph G = (V; E) to efficiently sample from the probability measure ¯G defined on the set of indepedent sets\Omega =\Omega G of G weight...

Consider the problem of approximately counting weighted independent sets of a graph G with activity λ, i.e., where the weight of an independent set I is λ |I |. We present a novel analysis yielding a deterministic approximation scheme which runs in polynomial time for any graph of maximum de-gree Δ and λ<λc =(Δ − 1) Δ−1 /(Δ − 2) Δ.Thisimproves on the previously known general bound of λ ≤ 2

7.2 was jointly undertaken with Vivek Gore, and is published here for the first time. I also thank an anonymous referee for carefully reading and providing helpful comments on a draft of this chapter. 1. Introduction The classical Monte Carlo method is an approach to estimating quantities that are hard to compute exactly. The quantity z of interest is expressed as the expectation z = ExpZ of a random variable (r.v.) Z for which some efficient sampling procedure is available. By taking the mean of some sufficiently large set of independent samples of Z, one may obtain an approximation to z. For example, suppose S = \Phi (x; y) 2 [0; 1] 2 : p i (x; y) 0; for all i \Psi<F12

We study discrete time Glauber dynamics for random configurations with local constraints (e.g. proper coloring, Ising and Potts models) on finite graphs with n vertices and of bounded degree. We show that the relaxation time (defined as the reciprocal of the spectral gap 1 \Gamma 2 ) for the dynamics on trees and on certain hyperbolic graphs, is polynomial in n. For these hyperbolic graphs, this yields a general polynomial sampling algorithm for random configurations. We then show that if the relaxation time 2 satisfies 2 = O(n), then the correlation coefficient, and the mutual information, between any local function (which depends only on the configuration in a fixed window) and the boundary conditions, decays exponentially in the distance between the window and the boundary. For the Ising model on a regular tree, this condition is sharp. 1.

Summary Approximate sampling from combinatorially-defined sets, using the Markov chain Monte Carlo method, is discussed from the perspective of combinatorial algorithms. We also examine the associated problem of discrete integration over such sets. Recent work is reviewed, and we re-examine the underlying formal foundational framework in the light of this. We give a detailed treatment of the coupling technique, a classical method for analysing the convergence rates of Markov chains. The related topic of perfect sampling is examined. In perfect sampling, the goal is to sample exactly from the target set. We conclude with a discussion of negative results in this area. These are results which imply that there are no polynomial time algorithms of a particular type for a particular problem. 1

This note considers the problem of sampling from the set of weighted independent sets of a graph with maximum degree #. For a positive fugacity #,theweight of an independent set # is # |#| . Luby and Vigoda proved that the Glauber dynamics, which only changes the configuration at a randomly chosen vertex in each step, has mixing time O(n log n)when#< 2 #-2 for triangle-free graphs. We extend their approach to general graphs. 1

Abstract. A new method for analyzing the mixing time of Markov chains is described. This method is an extension of path coupling and involves analyzing the coupling over multiple steps. The expected behavior of the coupling at a certain stopping time is used to bound the expected behavior of the coupling after a fixed number of steps. The new method is applied to analyze the mixing time of the Glauber dynamics for graph colorings. We show that the Glauber dynamics has O(n log(n)) mixing time for triangle-free ∆-regular graphs if k colors are used, where k ≥ (2 − η)∆, for some small positive constant η. This is the first proof of an optimal upper bound for the mixing time of the Glauber dynamics for some values of k in the range k ≤ 2∆.

We consider approximate counting of colourings of an n-vertex graph using rapidly mixing Markov chains. It has been shown by Jerrum and by Salas and Sokal that a simple random walk on graph colourings would mix rapidly provided the number of colours, k, exceeded the maximum degree \Delta of the graph by a factor of at least 2. Lack of improvements on this bound led to the conjecture that k 2\Delta was a natural barrier. We disprove this conjecture in the simplest case of 5-colouring graphs of maximum degree 3. Our proof involves a computer-assisted proof technique to establish rapid mixing of a new "heat bath" Markov chain on colourings using the method of path coupling. We outline an extension to 7-colourings of triangle-free 4-regular graphs. Since rapid mixing implies approximate counting in polynomial time, we show in contrast that exact counting is unlikly to be possible (in polynomial time). We give a general proof that the problem of exactly counting the number of proper k-colo...