Polynomial-time perfect matchings in dense hypergraphs

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 Karpi\'nski, Ruci\'nski and Szyma\'nska; Szyma\'nska 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.