The decision problem for perfect matchings in dense hypergraphs
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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 δn-ℓk-ℓ. 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.
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