On the Keevash-Knox-Mycroft conjecture

Given $1\le \ell <k$ and $\delta \ge 0$, let $\text{PM}(k,\ell ,\delta )$ be the decision problem for the existence of perfect matchings in n-vertex k-uniform hypergraphs with minimum ℓ-degree at least $\delta \left(\begin{array}{c}n-\ell \\ k-\ell \end{array}\right)$. For $k\ge 3$, $\text{PM}(k,\ell ,0)$ was one of the first NP-complete problems by Karp. Keevash, Knox and Mycroft conjectured that $\text{PM}(k,\ell ,\delta )$ is in P for every $\delta >1-{(1-1/k)}^{k-\ell }$ and verified the case $\ell =k-1$.

In this paper we show that this problem can be reduced to the study of the minimum ℓ-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.