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

TL;DR

This survey compares techniques for bounding Markov chain mixing times, highlighting the strengths and limitations of conductance, canonical paths, and coupling methods, and discusses recent advances and open questions in the field.

Contribution

It provides a comprehensive comparison of conductance, canonical paths, and coupling techniques for proving rapid mixing, including new insights into their applicability and limitations.

Findings

Canonical paths can capture rapid mixing for many chains.

Path Coupling often yields simpler and tighter bounds.

Coupling cannot prove rapid mixing for all chains, such as the Jerrum-Sinclair chain.

Abstract

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.