The Decision Problem for Perfect Matchings in Dense Hypergraphs

Given 1 ≤ ℓ < k and δ ≥ 0, let PM(k, ℓ, δ) be the decision problem for the existence of perfect matchings in n-vertex k-uniform hypergraphs with minimum ℓ-degree at least (Equation presented). For k ≥ 3, the decision problem in general k-uniform hypergraphs, equivalently PM(k, ℓ, 0), is one of Karp's 21 NP-complete problems. Moreover, for k ≥ 3, a reduction of Szymańska showed that PM(k, ℓ, δ) is NP-complete for δ < 1 − (1 − 1/k)^k−ℓ. A breakthrough by Keevash, Knox and Mycroft [STOC'13] resolved this problem for ℓ = k − 1 by showing that PM(k, k − 1, δ) is in P for δ > 1/k. Based on their result for ℓ = k − 1, Keevash, Knox and Mycroft conjectured that PM(k, ℓ, δ) is in P for every δ > 1 − (1 − 1/k)^k−ℓ. In this paper it is shown that this decision problem for perfect matchings can be reduced to the study of the minimum ℓ-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 ℓ ≥ 0.4k.