Finding matchings in dense hypergraphs

TL;DR

This paper presents a fixed parameter tractable algorithm for deciding the existence of large matchings in dense hypergraphs, which can efficiently find or certify the absence of such matchings.

Contribution

It introduces a fixed parameter tractable algorithm for matching decision problems in dense hypergraphs, improving computational efficiency for certain parameters.

Findings

Algorithm decides matching existence in dense hypergraphs

Algorithm finds a matching or certifies non-existence

Polynomial-time for specific parameters when m=n/k and c=O(log n)

Abstract

We consider the algorithmic decision problem that takes as input an $n$-vertex $k$-uniform hypergraph $H$ with minimum codegree at least $m-c$ and decides whether it has a matching of size $m$. We show that this decision problem is fixed parameter tractable with respect to $c$. Furthermore, our algorithm not only decides the problem, but actually either finds a matching of size $m$ or a certificate that no such matching exists. In particular, when $m=n/k$ and $c=O(\log n)$, this gives a polynomial-time algorithm, that given any $n$-vertex $k$-uniform hypergraph $H$ with minimum codegree at least $n/k-c$, finds either a perfect matching in $H$ or a certificate that no perfect matching exists.