Rapidly Mixing Markov Chains: A Comparison of Techniques (A Survey)

We survey existing techniques to bound the mixing time of Markov chains. The mixing time is related to a geometric parameter called conductance which is a measure of edge-expansion. Bounds on conductance are typically obtained by a technique called "canonical paths" where the idea is to find a set of paths, one between every source-destination pair, such that no edge is heavily congested. However, the canonical paths approach cannot always show rapid mixing of a rapidly mixing chain. This drawback disappears if we allow the flow between a pair of states to be spread along multiple paths. We prove that for a large class of Markov chains canonical paths does capture rapid mixing. Allowing multiple paths to route the flow still does help a great deal in proofs, as illustrated by a result of Morris & Sinclair (FOCS'99) on the rapid mixing of a Markov chain for sampling 0/1 knapsack solutions. A different approach to prove rapid mixing is "Coupling". Path Coupling is a variant discovered by Bubley & Dyer (FOCS'97) that often tremendously reduces the complexity of designing good Couplings. We present several applications of Path Coupling in proofs of rapid mixing. These invariably lead to much better bounds on mixing time than known using conductance, and moreover Coupling based proofs are typically simpler. This motivates the question of whether Coupling can be made to work whenever the chain is rapidly mixing. This question was answered in the negative by Kumar & Ramesh (FOCS'99), who showed that no Coupling strategy can prove the rapid mixing of the Jerrum-Sinclair chain for sampling perfect and near-perfect matchings.

[1]  Alistair Sinclair,et al.  Improved Bounds for Mixing Rates of Markov Chains and Multicommodity Flow , 1992, Combinatorics, Probability and Computing.

[2]  Mark Jerrum,et al.  Polynomial-Time Approximation Algorithms for the Ising Model , 1990, SIAM J. Comput..

[3]  Martin Dyer,et al.  A more rapidly mixing Markov chain for graph colorings , 1998 .

[4]  Martin E. Dyer,et al.  A Mildly Exponential Time Algorithm for Approximating the Number of Solutions to a Multidimensional Knapsack Problem , 1993, Combinatorics, Probability and Computing.

[5]  Prasad Tetali,et al.  Simple Markov-chain algorithms for generating bipartite graphs and tournaments , 1997, SODA '97.

[6]  Martin E. Dyer,et al.  An elementary analysis of a procedure for sampling points in a convex body , 1998, Random Struct. Algorithms.

[7]  M. Habib.  Probabilistic methods for algorithmic discrete mathematics , 1998 .

[8]  Nabil Kahale,et al.  A semidefinite bound for mixing rates of Markov chains , 1996, Random Struct. Algorithms.

[9]  Mark Jerrum,et al.  Mathematical Foundations of the Markov Chain Monte Carlo Method , 1998 .

[10]  Mark Jerrum,et al.  Approximating the Permanent , 1989, SIAM J. Comput..

[11]  Alistair Sinclair,et al.  Random walks on truncated cubes and sampling 0-1 knapsack solutions , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[12]  DyerMartin,et al.  A random polynomial-time algorithm for approximating the volume of convex bodies , 1991 .

[13]  Tomás Feder,et al.  Balanced matroids , 1992, STOC '92.

[14]  L. Khachiyan,et al.  On the conductance of order Markov chains , 1991 .

[15]  Mark Jerrum,et al.  Approximate Counting, Uniform Generation and Rapidly Mixing Markov Chains , 1987, International Workshop on Graph-Theoretic Concepts in Computer Science.

[16]  Prabhakar Raghavan,et al.  Randomized rounding: A technique for provably good algorithms and algorithmic proofs , 1985, Comb..

[17]  Alistair Sinclair,et al.  Algorithms for Random Generation and Counting: A Markov Chain Approach , 1993, Progress in Theoretical Computer Science.

[18]  Stochastic Inequalities,et al.  HITTING-TIME AND OCCUPATION-TIME BOUNDS IMPLIED BY DRIFT ANALYSIS WITH APPLICATIONS , 1982 .

[19]  Martin E. Dyer,et al.  On Counting Independent Sets in Sparse Graphs , 2002, SIAM J. Comput..

[20]  Martin E. Dyer,et al.  On Markov Chains for Independent Sets , 2000, J. Algorithms.

[21]  P. Diaconis,et al.  Geometric Bounds for Eigenvalues of Markov Chains , 1991 .

[22]  Mark Jerrum,et al.  The Markov chain Monte Carlo method: an approach to approximate counting and integration , 1996 .

[23]  Martin Dyer,et al.  A Sub-exponential Time Algorithm for Approximating the Number of Solutions to a Multidimensional Knapsack Problem , 1993 .

[24]  Bojan Mohar,et al.  Isoperimetric numbers of graphs , 1989, J. Comb. Theory, Ser. B.

[25]  A more rapidly mixing Markov chain for graph colorings , 1998, Random Struct. Algorithms.

[26]  Milena Mihail,et al.  On Coupling and the Approximation of the Permanent , 1989, Inf. Process. Lett..

[27]  Martin E. Dyer,et al.  A Genuinely Polynomial-Time Algorithms for Sampling Two-Rowed Contingency Tables , 1998, ICALP.

[28]  Frank Thomson Leighton,et al.  An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

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

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

[31]  Mark Jerrum,et al.  Approximate Counting, Uniform Generation and Rapidly Mixing Markov Chains , 1987, WG.

[32]  Martin E. Dyer,et al.  Path coupling: A technique for proving rapid mixing in Markov chains , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[33]  P. Flajolet.  On approximate counting , 1982 .

[34]  Mark Jerrum,et al.  An elementary analysis of a procedure for sampling points in a convex body , 1998 .

[35]  M. Dyer.  Computing the volume of convex bodies : a case where randomness provably helps , 1991 .

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

[37]  D. Aldous.  Some Inequalities for Reversible Markov Chains , 1982 .

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

[39]  P. Matthews.  Generating a Random Linear Extension of a Partial Order , 1991 .

[40]  V. S. Anil Kumar,et al.  Coupling vs. conductance for the Jerrum-Sinclair chain , 2001, Random Struct. Algorithms.

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

[42]  Leslie G. Valiant,et al.  Random Generation of Combinatorial Structures from a Uniform Distribution , 1986, Theor. Comput. Sci..

[43]  Milena Mihail,et al.  Conductance and convergence of Markov chains-a combinatorial treatment of expanders , 1989, 30th Annual Symposium on Foundations of Computer Science.

[44]  Andrei Z. Broder,et al.  How hard is it to marry at random? (On the approximation of the permanent) , 1986, STOC '86.

[45]  Eric Vigoda,et al.  Improved bounds for sampling colorings , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).

[46]  Martin E. Dyer,et al.  On counting independent sets in sparse graphs , 1999, 40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039).