Computing the partition function for perfect matchings in a hypergraph

TL;DR

This paper introduces a polynomial-time approximation algorithm for computing the sum over all partitions of a hypergraph into k-subsets, extending classical bounds and applying to counting matchings.

Contribution

It presents a novel polynomial-time approximation method for the hypergraph partition sum, generalizing bounds for permanents and enabling counting of matchings.

Findings

Provides a polynomial-time approximation algorithm for the hypergraph partition sum.

Extends van der Waerden and Bregman-Minc bounds to higher dimensions.

Applies to counting perfect and nearly perfect matchings in hypergraphs.

Abstract

Given non-negative weights w_S on the k-subsets S of a km-element set V, we consider the sum of the products w_{S_1} ... w_{S_m} for all partitions V = S_1 cup ... cup S_m into pairwise disjoint k-subsets S_i. When the weights w_S 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.