The decision problem for perfect matchings in dense hypergraphs

Given $1\le \ell <k$ and $\delta>0$, let $\textbf{PM}(k,\ell,\delta)$ be the decision problem for the existence of perfect matchings in $n$-vertex $k$-uniform hypergraphs with minimum $\ell$-degree at least $\delta\binom{n-\ell}{k-\ell}$. For $k\ge 3$, the decision problem in general $k$-uniform hypergraphs, equivalently $\textbf{PM}(k,\ell,0)$, is one of Karp's 21 NP-complete problems. Moreover, a reduction of Szyma\'{n}ska showed that $PM(k, \ell, \delta)$ is NP-complete for $\delta < 1-(1-1/k)^{k-\ell}$. A breakthrough by Keevash, Knox and Mycroft [STOC '13] resolved this problem for $\ell=k-1$ by showing that $PM(k, k-1, \delta)$ is in P for $\delta > 1/k$. Based on their result for $\ell=k-1$, Keevash, Knox and Mycroft conjectured that $PM(k, \ell, \delta)$ is in P for every $\delta > 1-(1-1/k)^{k-\ell}$. In this paper it is shown that this decision problem for perfect matchings can be reduced to the study of the minimum $\ell$-degree condition forcing the existence of fractional perfect matchings. That is, we hopefully solve the "computational complexity" aspect of the problem by reducing it to a well-known extremal problem in hypergraph theory. In particular, together with existing results on fractional perfect matchings, this solves the conjecture of Keevash, Knox and Mycroft for $\ell\ge 0.4k$.