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 δ binom(n-𝓁,k-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.
@InProceedings{gan_et_al:LIPIcs.ICALP.2022.64, author = {Gan, Luyining and Han, Jie}, title = {{The Decision Problem for Perfect Matchings in Dense Hypergraphs}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {64:1--64:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.64}, URN = {urn:nbn:de:0030-drops-164057}, doi = {10.4230/LIPIcs.ICALP.2022.64}, annote = {Keywords: Computational Complexity, Perfect Matching, Hypergraph} }
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