Luyining Gan, Jie Han
Given $1\le \ell <k$ and $δ\ge0$, let $\textbf{PM}(k,\ell,δ)$ be the decision problem for the existence of perfect matchings in $n$-vertex $k$-uniform hypergraphs with minimum $\ell$-degree at least $δ\binom{n-\ell}{k-\ell}$. For $k\ge 3$, $\textbf{PM}(k,\ell,0)$ was one of the first NP-complete problems by Karp. Keevash, Knox and Mycroft conjectured that $\textbf{PM}(k, \ell, δ)$ is in P for every $δ> 1-(1-1/k)^{k-\ell}$ and verified the case $\ell=k-1$. In this paper we show that this problem can be reduced to the study of the minimum $\ell$-degree condition forcing the existence of fractional perfect matchings. Together with existing results on fractional perfect matchings, this solves the conjecture of Keevash, Knox and Mycroft for $\ell\ge 0.4k$. Moreover, we also supply an algorithm that outputs a perfect matching, provided that one exists.