Path coupling: A technique for proving rapid mixing in Markov chains

The main technique used in algorithm design for approximating #P-hard 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 applications 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. However the true power of the method is that the simplification 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 TWICE-SAT, and problems from statistical physics, such as the antiferromagnetic Potts model and the hard-core lattice gas model. In each case we provide either a proof of rapid mixing where none was known previously, or substantial simplification of existing proofs with consequent gains in the performance of the resulting algorithms.

[1]  Alan M. Frieze,et al.  Electronic Colloquium on Computational Complexity Polynomial Time Randomised Approximation Schemes for Tutte-grr Othendieck Invariants: the Dense Case , 2022 .

[2]  Martin E. Dyer,et al.  Beating the 2Δ bound for approximately counting colourings: a computer-assisted proof of rapid mixing , 1998, SODA '98.

[3]  Martin E. Dyer,et al.  A random polynomial-time algorithm for approximating the volume of convex bodies , 1991, JACM.

[4]  A. Sokal,et al.  Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem , 1996, cond-mat/9603068.

[5]  Dorit S. Hochbaum,et al.  Approximation Algorithms for NP-Hard Problems , 1996 .

[6]  Martin E. Dyer,et al.  Graph orientations with no sink and an approximation for a hard case of #SAT , 1997, SODA '97.

[7]  Claude Berge,et al.  Graphs and Hypergraphs , 2021, Clustering.

[8]  Richard M. Karp,et al.  Monte-Carlo algorithms for enumeration and reliability problems , 1983, 24th Annual Symposium on Foundations of Computer Science (sfcs 1983).

[9]  Eric Vigoda,et al.  Approximately counting up to four (extended abstract) , 1997, STOC '97.

[10]  Martin E. Dyer,et al.  Faster random generation of linear extensions , 1999, SODA '98.

[11]  D. Aldous Random walks on finite groups and rapidly mixing markov chains , 1983 .

[12]  Leslie G. Valiant,et al.  The Complexity of Enumeration and Reliability Problems , 1979, SIAM J. Comput..

[13]  Mark Jerrum,et al.  A Very Simple Algorithm for Estimating the Number of k-Colorings of a Low-Degree Graph , 1995, Random Struct. Algorithms.

[14]  M. Dyer,et al.  Path Coupling, Dobrushin Uniqueness, and Approximate Counting , 1997 .

[15]  Martin Dyer,et al.  A New Approach to Polynomial-Time Generation of Random Points in Convex Bodies , 1996 .