On the switch Markov chain for perfect matchings

TL;DR

This paper analyzes the switch Markov chain for perfect matchings in bipartite graphs, establishing conditions for ergodicity and polynomial mixing time, especially for monotone graphs relevant in statistics.

Contribution

It provides the first polynomial mixing time bounds for the switch chain on monotone graphs, advancing understanding of its efficiency in sampling perfect matchings.

Findings

The switch chain is ergodic for certain classes of bipartite graphs.

Polynomial mixing time is proven for monotone graphs.

Results are relevant for statistical sampling applications.

Abstract

We study a simple Markov chain, the switch chain, on the set of all perfect matchings in a bipartite graph. This Markov chain was proposed by Diaconis, Graham and Holmes as a possible approach to a sampling problem arising in Statistics. We ask: for which classes of graphs is the Markov chain ergodic and for which is it rapidly mixing? We provide a precise answer to the ergodicity question and close bounds on the mixing question. We show for the first time that the mixing time of the switch chain is polynomial in the case of monotone graphs, a class that includes examples of interest in the statistical setting.