Perfect Matchings in Random Sparsifications of Dense Hypergraphs

TL;DR

This work addresses robustness of the perfect matching problem in dense hypergraphs under random sparsification. By combining a strengthened partition-lattice-absorption framework with the spread method, the authors show that for dense $k$-graphs with $\delta_{k-1}(H) \ge n/k + \gamma n$, a polynomial-time algorithm can a.a.s. decide the presence of a perfect matching in the random subgraph $H_p$ when $p \ge C \log n / n^{k-1}$, and they obtain bounds on the number of perfect matchings. They extend the approach to $F$-factors in graphs and develop a general structural theorem ensuring a $q$-spread distribution on $F$-factors under divisibility and reachability conditions, leading to robust appearance in $H_p$ at thresholds of the form $p \ge K_{FKNP} C'' \log n / n^{1/m_1(F)}$, with refinements for strictly $1$-balanced $F$. The random clustering technique and lattice-based absorption underpin both algorithmic feasibility and probabilistic robustness, yielding not only decision results but also counting estimates for PMs and $F$-factors in random sparsifications. The results advance robust versions of Dirac-type problems in hypergraphs and highlight the effectiveness of spread-based methods in algorithmic and probabilistic combinatorics.

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, Müyesser and Pokrovskiy, based on the spread method.

Theorems & Definitions (69)

• Theorem 1.1: keevash2013polynomial
• Theorem 1.2: Main result
• Theorem 1.3: alon1996h
• Theorem 1.4: shokoufandeh2003proof
• Theorem 1.5: han2020complexity
• Definition 1.6: Partition, index vector and lattice
• Lemma 1.7: Partition lemma for $F$-factors
• Theorem 1.8: Robustness of $F$-factors
• Definition 1.9
• Theorem 1.10: Main technical result