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.

翻译：给定 $1\le \ell <k$ 和 $δ\ge0$，设 $\textbf{PM}(k,\ell,δ)$ 为判断 $n$ 顶点 $k$ 一致超图中是否存在完美匹配的决策问题，其中最小 $\ell$ 度至少为 $δ\binom{n-\ell}{k-\ell}$。对于 $k\ge 3$，$\textbf{PM}(k,\ell,0)$ 是Karp提出的首批NP完全问题之一。Keevash、Knox和Mycroft猜想：对所有 $δ> 1-(1-1/k)^{k-\ell}$，$\textbf{PM}(k, \ell, δ)$ 属于P类，并验证了 $\ell=k-1$ 的情形。本文证明该问题可归约为研究迫使分数完美匹配存在的最小 $\ell$ 度条件。结合现有关于分数完美匹配的结果，这解决了Keevash、Knox和Mycroft对 $\ell\ge 0.4k$ 的猜想。此外，我们还提供了一个算法，在完美匹配存在时能输出该匹配。