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54
Path coupling: A technique for proving rapid mixing in markov chains
 In FOCS ’97: Proceedings of the 38th Annual Symposium on Foundations of Computer Science (FOCS
, 1997
"... The main technique used in algorithm design for approximating #Phard counting problems is the Markov chain Monte Carlo method. At the heart of the method is the study of the convergence (mixing) rates of particular Markov chains of interest. In this paper we illustrate a new approach to the couplin ..."
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Cited by 150 (20 self)
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The main technique used in algorithm design for approximating #Phard counting problems is the Markov chain Monte Carlo method. At the heart of the method is the study of the convergence (mixing) rates of particular Markov chains of interest. In this paper we illustrate a new approach to the coupling technique, which we call path coupling, for bounding mixing rates. Previous appliccitions of coupling have required detailed insights into the combinatorics of the problem at hand, and this complexity can make the technique extremely difficult to apply successfully. Path coupling helps to minimize the combinatorial difficulty and in all cases provides simpler convergence proofs than does the standard coupling method. Howevel; the true power of the method i>i that the simpl$cation obtained may allow coupling proofs which were previously unknown, or provide significantly better bounds than those obtained using the standard method. We apply the path coupling method to several hard combinatorial problems, obtaining new or improved results. We examine combinatorial probr'ems such as graph colouring and TWICESAT, and problems fn?m statistical physics, such as the antiferromagnetic Potts model and the hardcore lattice gas model. In each case we provide either a proof of rapid mixing where none was known previously, or substantial simpl$cation of existing proofs with conseqent gains in the pegormance of the resulting algorithms. 1
On Counting Independent Sets in Sparse Graphs
, 1998
"... 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 ..."
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Cited by 62 (12 self)
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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 #Pcomplete, 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...
Counting independent sets up to the tree threshold
 In STOC ’06: Proceedings of the thirtyeighth annual ACM symposium on Theory of computing
, 2006
"... 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 degree Δ ..."
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Cited by 51 (2 self)
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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 degree Δ and λ<λc =(Δ − 1) Δ−1 /(Δ − 2) Δ.Thisimproves on the previously known general bound of λ ≤ 2
Fast Convergence of the Glauber Dynamics for Sampling Independent Sets: Part II
, 1999
"... 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 Glaube ..."
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Cited by 43 (4 self)
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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 trianglefree 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...
A more rapidly mixing Markov chain for graph colourings
, 1997
"... We define a new Markov chain on (proper) kcolourings 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/SalasSokal chain in most circumstances. For ..."
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Cited by 40 (11 self)
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We define a new Markov chain on (proper) kcolourings 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/SalasSokal 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...
Mathematical foundations of the Markov chain Monte Carlo method
 in Probabilistic Methods for Algorithmic Discrete Mathematics
, 1998
"... 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 a ..."
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Cited by 30 (1 self)
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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
Glauber Dynamics on Trees and Hyperbolic Graphs
, 2001
"... 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 dynami ..."
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Cited by 27 (11 self)
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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.
A Note on the Glauber Dynamics for Sampling Independent Sets
 Electronic Journal of Combinatorics
, 2001
"... 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 ..."
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Cited by 27 (3 self)
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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 trianglefree graphs. We extend their approach to general graphs. 1
Random walks on combinatorial objects
 Surveys in Combinatorics 1999
, 1999
"... Summary Approximate sampling from combinatoriallydefined 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 reexamine the unde ..."
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Cited by 25 (8 self)
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Summary Approximate sampling from combinatoriallydefined 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 reexamine 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
An extension of path coupling and its application to the Glauber dynamics for graph colorings
 SIAM Journal on Computing
, 2001
"... 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 cou ..."
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Cited by 21 (4 self)
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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 trianglefree ∆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∆.