Counting Perfect Matchings in Graphs of Degree 3

Abstract

Counting perfect matchings is an interesting and challenging combinatorial task. It has important applications in statistical physics. As the general problem is #P complete, it is usually tackled by randomized heuristics and approximation schemes. The trivial running times for exact algorithms are O ^*((n − 1)!!) = O ^*(n!!) = O ^*((n/2)! 2^n/2) for general graphs and O ^*((n/2)!) for bipartite graphs. Ryser’s old algorithm uses the inclusion exclusion principle to handle the bipartite case in time O ^*(2^n/2). It is still the fastest known algorithm handling arbitrary bipartite graphs.

For graphs with n vertices and m edges, we present a very simple argument for an algorithm running in time O ^*(1.4656^m − n). For graphs of average degree 3 this is O ^*(1.2106ⁿ), improving on the previously fastest algorithm of Björklund and Husfeldt. We also present an algorithm running in time O ^*(1.4205^m − n) or O ^*(1.1918ⁿ) for average degree 3 graphs. The purpose of these simple algorithms is to exhibit the power of the m − n measure.

Here, we don’t investigate the further improvements possible for larger average degrees by applying the measure-and-conquer method.