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URN: urn:nbn:de:0030-drops-164057
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The Decision Problem for Perfect Matchings in Dense Hypergraphs

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Abstract

Given 1 ≀ 𝓁 < k and Ξ΄ β‰₯ 0, let PM(k,𝓁,Ξ΄) be the decision problem for the existence of perfect matchings in n-vertex k-uniform hypergraphs with minimum 𝓁-degree at least Ξ΄ binom(n-𝓁,k-𝓁). For k β‰₯ 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 β‰₯ 3, a reduction of SzymaΕ„ska showed that PM(k, 𝓁, Ξ΄) is NP-complete for Ξ΄ < 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, Ξ΄) is in P for Ξ΄ > 1/k. Based on their result for 𝓁 = k-1, Keevash, Knox and Mycroft conjectured that PM(k, 𝓁, Ξ΄) is in P for every Ξ΄ > 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 β‰₯ 0.4k.

BibTeX - Entry

@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/opus/volltexte/2022/16405},
  URN =		{urn:nbn:de:0030-drops-164057},
  doi =		{10.4230/LIPIcs.ICALP.2022.64},
  annote =	{Keywords: Computational Complexity, Perfect Matching, Hypergraph}
}

Keywords: Computational Complexity, Perfect Matching, Hypergraph
Seminar: 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)
Issue date: 2022
Date of publication: 28.06.2022


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