Abstract. We present a polynomial-time randomized algorithm for estimating the permanent of an arbitrary n ×n matrix with nonnegative entries. This algorithm—technically a “fully-polynomial randomized approximation scheme”—computes an approximation that is, with high probability, within arbitrarily small specified relative error of the true value of the permanent. Categories and Subject Descriptors: F.2.2 [Analysis of algorithms and problem complexity]: Nonnumerical

In the area of statistical physics, Monte Carlo algorithms based on Markov chain simulation have been in use for many years. The validity of these algorithms depends crucially on the rate of convergence to equilibrium of the Markov chain being simulated. Unfortunately, the classical theory of stochastic processes hardly touches on the sort of non-asymptotic analysis required in this application. As a consequence, it had previously not been possible to make useful, mathematically rigorous statements about the quality of the estimates obtained. Within the last ten years, analytical tools have been devised with the aim of correcting this deficiency. As well as permitting the analysis of Monte Carlo algorithms for classical problems in statistical physics, the introduction of these tools has spurred the development of new approximation algorithms for a wider class of problems in combinatorial enumeration and optimization. The “Markov chain Monte Carlo ” method has been applied to a variety of such problems, and often provides the only known efficient (i.e., polynomial time) solution technique.

We give a O( log n)-approximation algorithm for sparsest cut, balanced separator, and graph conductance problems. This improves the O(log n)-approximation of Leighton and Rao (1988). We use a well-known semidefinite relaxation with triangle inequality constraints. Central to our analysis is a geometric theorem about projections of point sets in , whose proof makes essential use of a phenomenon called measure concentration.

This paper surveys various results about Markov chains on general (non-countable) state spaces. It begins with an introduction to Markov chain Monte Carlo (MCMC) algorithms, which provide the motivation and context for the theory which follows. Then, sufficient conditions for geometric and uniform ergodicity are presented, along with quantitative bounds on the rate of convergence to stationarity. Many of these results are proved using direct coupling constructions based on minorisation and drift conditions. Necessary and sufficient conditions for Central Limit Theorems (CLTs) are also presented, in some cases proved via the Poisson Equation or direct regeneration constructions. Finally, optimal scaling and weak convergence results for Metropolis-Hastings algorithms are discussed. None of the results presented is new, though many of the proofs are. We also describe some Open Problems.

We introduce the notion of "balance", and say that a matroid is balanced if the matroid and all its minors satisfy the property that, for a randomly chosen basis, the presence of an element can only make any other element less likely. We establish strong expansion properties for the bases-exchange graph of balanced matroids; consequently, the set of bases of a balanced matroid can be sampled and approximately counted using rapidly mixing Markov chains. Thus, the general problem of approximately counting bases (known to be #P-complete) is reduced to that of showing balance. Specific classes for which balance is known to hold include graphic and regular matroids.

. We develop a new method, based on renormalization group ideas (block decimation procedure), to prove, under an assumption of strong mixing in a finite cube o , a Logarithmic Sobolev Inequality for the Gibbs state of a discrete spin system. As a consequence we derive the hypercontractivity of the Markov semigroup of the associated Glauber dynamics and the exponential convergence to equilibrium in the uniform norm in all volumes "multiples" of the cube o . Work partially supported by grant SC1-CT91-0695 of the Commission of European Communities 25=aprile=1997 [1] 0:1 Section 1. Preliminaries, Definitions and Results In this paper we analyze the problem of the approach to equilibrium for a general, not necessarily ferromagnetic, Glauber dynamics, i.e. a single spin flip stochastic dynamics reversible with respect to the Gibbs measure of a classical discrete spin system with finite range, translation invariant interaction. We prove that, if the Gibbs measure satisfies a Strong Mix...

.<F3.827e+05> We consider a finite random walk on a weighted graph<F3.539e+05><F3.827e+05> G; we show that the fraction of time spent in a set of vertices<F3.539e+05> A<F3.827e+05> converges to the stationary probability<F3.539e+05><F3.827e+05><F3.539e+05><F3.827e+05> #(A) with error probability exponentially small in the length of the random walk and the square of the size of the deviation from<F3.539e+05><F3.827e+05><F3.539e+05><F3.827e+05> #(A). The exponential bound is in terms of the expansion of<F3.539e+05> G<F3.827e+05> and improves previous results of [D. Aldous,<F3.405e+05> Probab. Engrg. Inform.<F3.827e+05> Sci., 1 (1987), pp. 33--46], [L. Lovasz and M. Simonovits,<F3.405e+05> Random Structures<F3.827e+05> Algorithms, 4 (1993), pp. 359--412], [M. Ajtai, J. Komlos, and E. Szemeredi,<F3.405e+05> Deterministic simulation of<F3.827e+05> logspace, in Proc. 19th ACM Symp. on Theory of Computing, 1987]. We show that taking the sample average from one trajectory gives a more e#cien...

Random independent sets in graphs arise, for example, in statistical physics, in the hard-core model of a gas. A new rapidly mixing Markov chain for independent sets is defined in this paper. We show that it is rapidly mixing for a wider range of values of the parameter than the Luby-Vigoda chain, the best previously known. Moreover the new chain is apparently more rapidly mixing than the Luby-Vigoda chain for larger values of (unless the maximum degree of the graph is 4). An extension of the chain to independent sets in hypergraphs is described. This chain gives an efficient method for approximately counting the number of independent sets of hypergraphs with maximum degree two, or with maximum degree three and maximum edge size three. Finally, we describe a method which allows one, under certain circumstances, to deduce the rapid mixing of one Markov chain from the rapid mixing of another, with the same state space and stationary distribution. This method is applied to two Markov ch...

A popular technique for studying random properties of a combinatorial set is to design a Markov chain Monte Carlo algorithm. For many problems there are natural Markov chains connecting the set of allowable configurations which are based on local moves, or "Glauber dynamics." Typically these single site update algorithms are difficult to analyze, so often the Markov chain is modified to update several sites simultaneously. Recently there has been progress in analyzing these more complicated algorithms for several important combinatorial problems. In this work we use the comparison technique of Diaconis and Saloff-Coste to show that several of the natural single point update algorithms are efficient. The strategy is to relate the mixing rate of these algorithms to the corresponding non-local algorithms which have already been analyzed. This allows us to give polynomial bounds for single point update algorithms for problems such as generating planar tilings and random triangulations of c...

This paper outlines the use of rapidly mixing Markov Chains in randomized polynomial time algorithms to solve approximately certain counting prob-lems. They fall into two classes: combinatorial problems like counting the number of perfect matchings in certain graphs and geometric ones like computing the volumes of convex sets.