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The Decision Problem for Perfect Matchings in Dense Hypergraphs

Authors Luyining Gan , Jie Han

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Author Details

Luyining Gan
  • Department of Mathematics and Statistics, University of Nevada, Reno, NV, USA
Jie Han
  • School of Mathematics and Statistics and Center for Applied Math, Beijing Institute of Technology, China

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Luyining Gan and Jie Han. The Decision Problem for Perfect Matchings in Dense Hypergraphs. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 64:1-64:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Computational Complexity
  • Perfect Matching
  • Hypergraph


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