Perfect Matchings in Random Sparsifications of Dense Hypergraphs

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 $\epsilon >0$, such decision problem restricted to $k$-uniform hypergraphs $H$ satisfying that every $(k-1)$-set of vertices is in at least $(1\mathrm{/}k+\epsilon )\mathrm{\mid }H\mathrm{\mid }$ edges is tractable, and the quantity $1\mathrm{/}k$ 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.