Polynomial-time perfect matchings in dense hypergraphs

TL;DR

This paper presents a polynomial-time algorithm to find perfect matchings in dense hypergraphs with high minimum codegree, solving a problem previously known to be NP-hard at lower densities.

Contribution

It introduces a novel polynomial-time algorithm for perfect matchings in dense hypergraphs and characterizes hypergraphs without perfect matchings using lattice-based constructions.

Findings

Algorithm finds perfect matchings or certifies none exist in polynomial time.

Successfully characterizes hypergraphs with no perfect matching using lattice methods.

Addresses a previously NP-hard problem in hypergraph theory.

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 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.