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)! 2n/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*(2n/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*(1.4656m−n). For graphs of average degree 3 this is O*(1.2106n), improving on the previously fastest algorithm of Bjorklund and Husfeldt. We also present an algorithm running in time O*(1.4205m−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.