On Counting Perfect Matchings in General Graphs

TL;DR

This paper investigates the complexity of counting perfect matchings in general graphs, demonstrating the limitations of existing Markov chain methods and proposing a new FPRAS for certain graph classes.

Contribution

It proves the JSV chain is torpid mixing in general graphs and introduces a new FPRAS for graphs close to bipartite using Gallai-Edmonds decomposition.

Findings

JSV chain is torpid mixing in general graphs

New FPRAS for graphs near bipartite

Fixed-parameter tractable algorithm based on graph decomposition

Abstract

Counting perfect matchings has played a central role in the theory of counting problems. The permanent, corresponding to bipartite graphs, was shown to be #P-complete to compute exactly by Valiant (1979), and a fully polynomial randomized approximation scheme (FPRAS) was presented by Jerrum, Sinclair, and Vigoda (2004) using a Markov chain Monte Carlo (MCMC) approach. However, it has remained an open question whether there exists an FPRAS for counting perfect matchings in general graphs. In fact, it was unresolved whether the same Markov chain defined by JSV is rapidly mixing in general. In this paper, we show that it is not. We prove torpid mixing for any weighting scheme on hole patterns in the JSV chain. As a first step toward overcoming this obstacle, we introduce a new algorithm for counting matchings based on the Gallai-Edmonds decomposition of a graph, and give an FPRAS for counting matchings in graphs that are sufficiently close to bipartite. In particular, we obtain a fixed-parameter tractable algorithm for counting matchings in general graphs, parameterized by the greatest "order" of a factor-critical subgraph.