Rapid mixing for the hardcore Glauber dynamics and other Markov chains in bounded-treewidth graphs

TL;DR

This paper proves rapid mixing of Glauber dynamics and related Markov chains for sampling independent sets, dominating sets, and matchings in graphs with bounded treewidth, offering simpler algorithms for these problems.

Contribution

It introduces a divide-and-conquer framework for proving rapid mixing in bounded-treewidth graphs, extending previous methods and applying to various Markov chains.

Findings

Rapid mixing for Glauber dynamics on independent sets in bounded-treewidth graphs

Rapid mixing for chains on dominating sets, b-edge covers, and b-matchings

Simpler algorithms for sampling and approximate counting in these graph classes

Abstract

We give a new rapid mixing result for a natural random walk on the independent sets of a graph $G$. We show that when $G$ has bounded treewidth, this random walk -- known as the Glauber dynamics for the hardcore model -- mixes rapidly for all fixed values of the standard parameter $\lambda > 0$, giving a simple alternative to existing sampling algorithms for these structures. We also show rapid mixing for analogous Markov chains on dominating sets, $b$-edge covers, $b$-matchings, maximal independent sets, and maximal $b$-matchings. (For $b$-matchings, maximal independent sets, and maximal $b$-matchings we also require bounded degree.) Our results imply simpler alternatives to known algorithms for the sampling and approximate counting problems in these graphs. We prove our results by applying a divide-and-conquer framework we developed in a previous paper, as an alternative to the projection-restriction technique introduced by Jerrum, Son, Tetali, and Vigoda. We extend this prior framework to handle chains for which the application of that framework is not straightforward, strengthening existing results by Dyer, Goldberg, and Jerrum and by Heinrich for the Glauber dynamics on $q$-colorings of graphs of bounded treewidth and bounded degree.