A determinant-based algorithm for counting perfect matchings in a general graph

Authors:
Steve Chien
Affiliations:
University of California, Berkeley, CA
Venue:
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Year:
2004

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Abstract

We present a simple estimator for the number of perfect matchings in a general (non-bipartite) graph. Our estimator requires O(ε-23n/2) trials to obtain a (1 ± ε)-approximation of the correct value with high probability on a graph with 2n vertices in the worst case, and only a polynomial number (O(ε-2nω(n)) of trials on random graphs, where ω(n) is any function tending to infinity.Our algorithm is based on the following idea: For any graph G, construct its associated Tutte matrix T, and derive a random matrix B from it by replacing each variable in T with ±1 uniformly at random; then output det B. This estimator is a natural generalization of the Godsil--Gutman estimator for matchings in a bipartite graph, and our analysis of its performance on random graphs borrows generously from Frieze and Jerrum's analysis of a similar estimator for bipartite graphs.