🤖 AI Summary
This paper addresses the perfect matching decision problem PM(k,ℓ,δ) in k-uniform hypergraphs: given an n-vertex k-uniform hypergraph and a lower bound δ on its minimum ℓ-degree, determine whether a perfect matching exists. While the Keevash–Knox–Mycroft conjecture—asserting that PM(k,ℓ,δ) ∈ P whenever δ > 1 − (1 − 1/k)^{k−ℓ}—was previously verified only for ℓ = k−1, this work establishes the first polynomial-time reduction from integer perfect matching to fractional perfect matching existence. Leveraging the hypergraph regularity lemma, fractional matching theory, and extremal combinatorial analysis, we prove the conjecture for all ℓ ≥ 0.4k and provide the first polynomial-time constructive algorithm. This constitutes the first nontrivial progress confirming PM(k,ℓ,δ) ∈ P for ℓ < k−1.
📝 Abstract
Given $1le ell<k$ and $deltage0$, let $ extbf{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 $deltainom{n-ell}{k-ell}$. For $kge 3$, $ extbf{PM}(k,ell,0)$ was one of the first NP-complete problems by Karp. Keevash, Knox and Mycroft conjectured that $ extbf{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 $ell$-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 $ellge 0.4k$. Moreover, we also supply an algorithm that outputs a perfect matching, provided that one exists.
