Computing the partition function for perfect matchings in a hypergraph

Authors:
Alexander Barvinok;Alex Samorodnitsky
Affiliations:
Department of mathematics, university of michigan, ann arbor, mi 48109-1043, usa (e-mail: barvinok@umich.edu);Department of computer science, hebrew university of jerusalem, givat ram campus, 91904, israel (e-mail: salex@cs.huji.ac.il)
Venue:
Combinatorics, Probability and Computing
Year:
2011

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Abstract

Given non-negative weights wS on the k-subsets S of a km-element set V, we consider the sum of the products wS1 â聥聟â聥聟â聥聟 wSm over all partitions V = S1 â聢陋 â聥聟â聥聟â聥聟 â聢陋Sm into pairwise disjoint k-subsets Si. When the weights wS are 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 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.