Computing the Partition Function for Perfect Matchings in a Hypergraph

Given non-negative weights wS on the k-subsets S of a km-element set V , we consider the sum of the products wS1 · · ·wSm over all partitions V = S1 ∪ . . . ∪ Sm into pairwise disjoint k-subsets Si. When the weights wS are positive and within a constant factor, fixed in advance, of each other, we present a simple polynomial time algorithm to approximate the sum within a polynomial in m factor. In the process, we obtain higher-dimensional versions of the van der Waerden and Bregman-Minc bounds for permanents. We also discuss applications to counting of perfect and nearly perfect matchings in hypergraphs.