Results 1  10
of
66
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 ..."
Abstract

Cited by 175 (20 self)
 Add to MetaCart
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$cation obtained may allow coupling proofs which were previously unknown, or provide significantly better bounds than those obtained using the standard method. We apply the path coupling method to several hard combinatorial problems, obtaining new or improved results. We examine combinatorial problems such as graph colouring and TWICESAT, and problems fn?m statistical physics, such as the antiferromagnetic Potts model and the hardcore lattice gas model. In each case we provide either a proof of rapid mixing where none was known previously, or substantial simpl$cation of existing proofs with conseqent gains in the pegormance of the resulting algorithms.
On Markov chains for independent sets
 Journal of Algorithms
, 1997
"... Random independent sets in graphs arise, for example, in statistical physics, in the hardcore 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 LubyVigoda chain, ..."
Abstract

Cited by 82 (17 self)
 Add to MetaCart
(Show Context)
Random independent sets in graphs arise, for example, in statistical physics, in the hardcore 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 LubyVigoda chain, the best previously known. Moreover the new chain is apparently more rapidly mixing than the LubyVigoda 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...
Bounds On The Complex Zeros Of (Di)Chromatic Polynomials And PottsModel Partition Functions
 Chromatic Roots Are Dense In The Whole Complex Plane, Combinatorics, Probability and Computing
"... I show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc q  < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the ..."
Abstract

Cited by 61 (14 self)
 Add to MetaCart
(Show Context)
I show that there exist universal constants C(r) < ∞ such that, for all loopless graphs G of maximum degree ≤ r, the zeros (real or complex) of the chromatic polynomial PG(q) lie in the disc q  < C(r). Furthermore, C(r) ≤ 7.963907r. This result is a corollary of a more general result on the zeros of the Pottsmodel partition function ZG(q, {ve}) in the complex antiferromagnetic regime 1 + ve  ≤ 1. The proof is based on a transformation of the Whitney–Tutte–Fortuin–Kasteleyn representation of ZG(q, {ve}) to a polymer gas, followed by verification of the Dobrushin–Koteck´y–Preiss condition for nonvanishing of a polymermodel partition function. I also show that, for all loopless graphs G of secondlargest degree ≤ r, the zeros of PG(q) lie in the disc q  < C(r) + 1. KEY WORDS: Graph, maximum degree, secondlargest degree, chromatic polynomial,
The SwendsenWang process does not always mix rapidly
 Proc. 29th ACM Symp. on Theory of Computing
, 1997
"... The SwendsenWang process provides one possible dynamics for the Qstate Potts model in statistical physics. Computer simulations of this process are widely used to estimate the expectations of various observables (random variables) of a Potts system in the equilibrium (or Gibbs) distribution. The l ..."
Abstract

Cited by 51 (4 self)
 Add to MetaCart
(Show Context)
The SwendsenWang process provides one possible dynamics for the Qstate Potts model in statistical physics. Computer simulations of this process are widely used to estimate the expectations of various observables (random variables) of a Potts system in the equilibrium (or Gibbs) distribution. The legitimacy of such simulations depends on the rate of convergence of the process to equilibrium, often known as the mixing rate. Empirical observations suggest that the SwendsenWang process mixes rapidly in many instances of practical interest. In spite of this, we show that there are occasions on which the SwendsenWang process requires exponential time (in the size of the system) to approach equilibrium.
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 ..."
Abstract

Cited by 46 (12 self)
 Add to MetaCart
(Show Context)
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...
Exact Sampling and Approximate Counting Techniques
"... We present two algorithms for uniformly sampling from the proper colorings of a graph using k colors. We use exact sampling from the stationary distribution of a Markov chain with states that are the kcolorings of a graph with maximum degree ¢. As opposed to approximate sampling algorithms based on ..."
Abstract

Cited by 35 (10 self)
 Add to MetaCart
(Show Context)
We present two algorithms for uniformly sampling from the proper colorings of a graph using k colors. We use exact sampling from the stationary distribution of a Markov chain with states that are the kcolorings of a graph with maximum degree ¢. As opposed to approximate sampling algorithms based on rapid mixing, these algorithms have termination criteria that allow them to stop on some inputs much more quickly than in the worst case running time bound. For the first algorithm we show that when ¡¤£, the algorithm has an upper limit on the expected running time that is polynomial. For the second algorithm we show that for ¡�£�� where ¢� � � is an integer that satisfies ����£�� the running time is polynomial. Previously, Jerrum showed that it was possible to approximately sample uniformly in polynomial time from the set of ¡colorings when ¡�� but our algorithm is the first polynomial time exact sampling algorithm for this problem. Using approximate sampling, Jerrum also showed how to approximately count the number of ¡colorings. We give a new procedure for approximately counting the number of ¡colorings that improves the running time of the procedure of Jerrum by a factor of is the number of nodes in �� � when ¥����, where �� ¢ � the graph to be colored � and is the number of edges. In addition, we present an improved analysis of the chain of Luby and Vigoda for exact sampling from the independent sets of a graph. Finally, we present the first polynomial time method for exactly sampling from the sink free orientations of a graph. Bubley and Dyer showed how to approximately sample from this state space ��¥���������¥��������� � in time, our algorithm ��¥��¦�� � takes expected time.
A NonMarkovian Coupling for Randomly Sampling Colorings
, 2005
"... We study a simple Markov chain, known as the Glauber dynamics, for randomly sampling (proper) kcolorings of an input graph G on n vertices with maximum degree ∆ and girth g. We prove the Glauber dynamics is close to the uniform distribution after O(n log n) steps whenever k> (1 + ɛ)∆, for all ɛ& ..."
Abstract

Cited by 33 (6 self)
 Add to MetaCart
We study a simple Markov chain, known as the Glauber dynamics, for randomly sampling (proper) kcolorings of an input graph G on n vertices with maximum degree ∆ and girth g. We prove the Glauber dynamics is close to the uniform distribution after O(n log n) steps whenever k> (1 + ɛ)∆, for all ɛ> 0, assuming g ≥ 11 and ∆ = Ω(log n). The best previously known bounds were k> 11∆/6 for general graphs, and k> 1.489 ∆ for graphs satisfying girth and maximum degree requirements. Our proof relies on the construction and analysis of a nonMarkovian coupling. This appears to be the first application of a nonMarkovian coupling to substantially improve upon known results.
Correlation decay and deterministic FPTAS for counting listcolorings of a graph
 In Proc. 18th ACMSIAM Symp. Discret. Algorithms (2007), SIAM
"... Abstract We propose a deterministic algorithm for approximately counting the number of list colorings of a graph. Under the assumption that the graph is triangle free, the size of every list is at least α∆, where α is an arbitrary constant bigger than α * * = 2.8432 . . ., the solution of αe − 1 α ..."
Abstract

Cited by 30 (9 self)
 Add to MetaCart
(Show Context)
Abstract We propose a deterministic algorithm for approximately counting the number of list colorings of a graph. Under the assumption that the graph is triangle free, the size of every list is at least α∆, where α is an arbitrary constant bigger than α * * = 2.8432 . . ., the solution of αe − 1 α = 2, and ∆ is the maximum degree of the graph, we obtain the following results. For the case when the size of the each list is a large constant, we show the existence of a deterministic FPTAS for computing the total number of list colorings. The same deterministic algorithm has complexity 2 O(log 2 n) , without any assumptions on the sizes of the lists, where n is the size of the instance. Our results are not based on the most powerful existing counting technique rapidly mixing Markov chain method. Rather we build upon concepts from statistical physics, in particular, the decay of correlation phenomena and its implication for the uniqueness of Gibbs measures in infinite graphs. This approach was proposed in two recent papers [BG06] and [Wei05]. The principle insight of the present work is that the correlation decay property can be established with respect to certain computation tree, as opposed to the conventional correlation decay property which is typically established with respect to graph theoretic neighborhoods of a given node. This allows truncation of computation at a logarithmic depth in order to obtain polynomial accuracy in polynomial time. While the analysis conducted in this paper is limited to the problem of counting list colorings, the proposed algorithm can be extended to an arbitrary constraint satisfaction problem in a straightforward way.
Combinatorial Criteria for Uniqueness of Gibbs Measures, Random Structures and Algorithms
, 2005
"... We generalize previously known conditions for uniqueness of the Gibbs measure in statistical physics models by presenting conditions of any finite size for models on any underlying graph. We give two dual conditions, one requiring that the total influence on a site is small, and the other that the t ..."
Abstract

Cited by 26 (2 self)
 Add to MetaCart
(Show Context)
We generalize previously known conditions for uniqueness of the Gibbs measure in statistical physics models by presenting conditions of any finite size for models on any underlying graph. We give two dual conditions, one requiring that the total influence on a site is small, and the other that the total influence of a site is small. Our proofs are combinatorial in nature and use tools from the analysis of discrete Markov chains, in particular the path coupling method. The implications of our conditions for the mixing time of natural Markov chains associated with the models are discussed as well. We also present some examples of models for which the conditions hold.
On exact simulation of Markov random fields using coupling from the past
, 1997
"... A general framework for exact simulation of Markov random fields using the ProppWilson coupling from the past approach is proposed. Our emphasis is on situations lacking the monotonicity properties that have been exploited in previous studies. A critical aspect is the convergence time of the algori ..."
Abstract

Cited by 23 (2 self)
 Add to MetaCart
A general framework for exact simulation of Markov random fields using the ProppWilson coupling from the past approach is proposed. Our emphasis is on situations lacking the monotonicity properties that have been exploited in previous studies. A critical aspect is the convergence time of the algorithm; this we study both theoretically and experimentically.