FPTAS for Counting Weighted Edge Covers

An edge cover of a graph is a set of edges in which each vertex has at least one of its incident edges. The problem of counting the number of edge covers is #P-complete and was shown to admit a fully polynomial-time approximation scheme (FPTAS) recently [10]. Counting weighted edge covers is the problem of computing the sum of the weights for all the edge covers, where the weight of each edge cover is defined to be the product of the edge weights of all the edges in the cover. The FPTAS in [10] cannot apply to general weighted counting for edge covers, which was stated as an open question there. Such weighted counting is generally interesting as for instance the weighted counting independent sets (vertex covers) problem has been exhaustively studied in both statistical physics and computer science. Weighted counting for edge cover is especially interesting as it is closely related to counting perfect matchings, which is a long-standing open question. In this paper, we obtain an FPTAS for counting general weighted edge covers, and thus solve an open question in [10]. Our algorithm also goes beyond that to certain generalization of edge cover.