Alternative Algorithms for Counting All Matchings in Graphs

Abstract

We present two new methods for counting all matchings in a graph. Both methods are alternatives to methods based on the Markov Chains and both are unbiased. The first one is a generalization of a Godman-Godsil estimator. We show that it works in time O(1.0878ⁿ∈^-2) for general graphs. For dense graphs (every vertex is connected with at least ( 1/2 +α)n other vertices) it works in time O(n⁴+^{(6 ln 6)/α∈-2), where n is the number of vertices of a given graph and 0 < ∈ < 1 is an expected relative error. We also analyze the efficiency of the presented algorithm applied for random graphs. The second method uses importance sampling. This method works in exponential time but it can easily be enriched in some heuristics leading to very efficient algorithms in practice. Experiments show that our methods give better estimates than the Markow Chain approach}