Exact Covers via Determinants

TL;DR

This paper introduces randomized polynomial space algorithms for solving the k-dimensional matching and exact cover by k-sets problems, significantly improving previous methods especially for small k, by embedding Lovasz' perfect matching detection into an inclusion-exclusion framework.

Contribution

It develops new randomized algorithms for hypergraph matching and exact cover problems using determinants and inclusion-exclusion, with improved exponential time bounds.

Findings

Algorithms run in time O*(2^{n(k-2)/k}) and O*(c_k^n) for small k.

Significant improvement over previous algorithms for small k.

Embedding Lovasz' perfect matching detection into inclusion-exclusion is a key innovation.

Abstract

Given a k-uniform hypergraph on n vertices, partitioned in k equal parts such that every hyperedge includes one vertex from each part, the k-dimensional matching problem asks whether there is a disjoint collection of the hyperedges which covers all vertices. We show it can be solved by a randomized polynomial space algorithm in time O*(2^(n(k-2)/k)). The O*() notation hides factors polynomial in n and k. When we drop the partition constraint and permit arbitrary hyperedges of cardinality k, we obtain the exact cover by k-sets problem. We show it can be solved by a randomized polynomial space algorithm in time O*(c_k^n), where c_3=1.496, c_4=1.642, c_5=1.721, and provide a general bound for larger k. Both results substantially improve on the previous best algorithms for these problems, especially for small k, and follow from the new observation that Lovasz' perfect matching detection via determinants (1979) admits an embedding in the recently proposed inclusion-exclusion counting scheme for set covers, despite its inability to count the perfect matchings.