Perfect Matchings in Random Sparsifications of Dense Hypergraphs

TL;DR

This paper investigates the existence of perfect matchings in random sparsifications of dense hypergraphs, providing polynomial-time algorithms and bounds, and extends results to the F-factor problem in graphs.

Contribution

It introduces new probabilistic bounds and algorithms for perfect matchings in sparsified hypergraphs, extending previous deterministic results to a randomized setting.

Findings

High-probability polynomial-time decision algorithms

Effective bounds on the number of perfect matchings

Extension of results to F-factor problems in graphs

Abstract

The decision problem of perfect matchings in uniform hypergraphs is famously an NP-complete problem. It has been shown by Keevash--Knox--Mycroft [STOC, 2013] that for every $\varepsilon>0$, such decision problem restricted to $k$-uniform hypergraphs $H$ satisfying that every $(k-1)$-set of vertices is in at least $(1/k+\varepsilon)|H|$ edges is tractable, and the quantity $1/k$ is best possible. In this paper we study the existence of perfect matchings in the random $p$-sparsification of such $k$-uniform hypergraphs, that is, for $p=p(n)\in [0,1]$, every edge is kept with probability $p$ independent of others. As a consequence, we give a polynomial-time algorithm that with high probability solves the decision problem; we also derive effective bounds on the number of perfect matchings in such hypergraphs. At last, similar results are obtained for the $F$-factor problem in graphs. The key ingredients of the proofs are a strengthened partition lemma for the lattice-based absorption method, and the random redistribution method developed recently by Kelly, M\"uyesser and Pokrovskiy, based on the spread method.