Computing the Partition Function for Perfect Matchings in a Hypergraph

Abstract

Given non-negative weightswSon thek-subsetsSof akm-element setV, we consider the sum of the productswS1⋅⋅⋅wSmover all partitionsV=S1∪ ⋅⋅⋅ ∪Sminto pairwise disjointk-subsetsSi. When the weightswSare positive and within a constant factor of each other, fixed in advance, we present a simple polynomial-time algorithm to approximate the sum within a polynomial inmfactor. 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.