Polynomial-time perfect matchings in dense hypergraphs

Authors:
Peter Keevash;Fiachra Knox;Richard Mycroft
Affiliations:
Queen Mary, University of London, London, United Kingdom;Queen Mary, University of London, London, United Kingdom;University of Birmingham, Birmingham, United Kingdom
Venue:
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Year:
2013

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Abstract

Let H be a k-graph on n vertices, with minimum codegree at least n/k + cn for some fixed c 0. In this paper we construct a polynomial-time algorithm which finds either a perfect matching in H or a certificate that none exists. This essentially solves a problem of Karpinski, Rucinski and Szymanska, who previously showed that this problem is NP-hard for a minimum codegree of n/k - cn. Our algorithm relies on a theoretical result of independent interest, in which we characterise any such hypergraph with no perfect matching using a family of lattice-based constructions.