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*(1.2106 n ), 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*(1.1918 n ) 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. © 2012 Springer-Verlag Berlin Heidelberg.