Approximate Counting of Matchings in Sparse Uniform Hypergraphs

TL;DR

This paper presents a fully polynomial randomized approximation scheme for counting matchings in certain sparse hypergraphs, extending canonical path methods and Euler tour techniques, while proving NP-hardness in more general cases.

Contribution

It introduces a novel FPRAS for matchings in hypergraphs with few claws, generalizing canonical path methods to hypergraphs with local restrictions.

Findings

FPRAS developed for hypergraphs with few claws

Extension of canonical path method to hypergraphs

NP-hardness proven for general hypergraphs without restrictions

Abstract

In this paper we give a fully polynomial randomized approximation scheme (FPRAS) for the number of matchings in k-uniform hypergraphs whose intersection graphs contain few claws. Our method gives a generalization of the canonical path method of Jerrum and Sinclair to hypergraphs satisfying a local restriction. Our proof method depends on an application of the Euler tour technique for the canonical paths of the underlying Markov chains. On the other hand, we prove that it is NP-hard to approximate the number of matchings even for the class of k-uniform, 2-regular and linear hypergraphs, for all k >= 6, without the above restriction.