eng
Schloss Dagstuhl β Leibniz-Zentrum fΓΌr Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-28
64:1
64:16
10.4230/LIPIcs.ICALP.2022.64
article
The Decision Problem for Perfect Matchings in Dense Hypergraphs
Gan, Luyining
1
https://orcid.org/0000-0002-4196-9761
Han, Jie
2
https://orcid.org/0000-0002-2013-2962
Department of Mathematics and Statistics, University of Nevada, Reno, NV, USA
School of Mathematics and Statistics and Center for Applied Math, Beijing Institute of Technology, China
Given 1 β€ π < k and Ξ΄ β₯ 0, let PM(k,π,Ξ΄) be the decision problem for the existence of perfect matchings in n-vertex k-uniform hypergraphs with minimum π-degree at least Ξ΄ binom(n-π,k-π). For k β₯ 3, the decision problem in general k-uniform hypergraphs, equivalently PM(k,π,0), is one of Karpβs 21 NP-complete problems. Moreover, for k β₯ 3, a reduction of SzymaΕska showed that PM(k, π, Ξ΄) is NP-complete for Ξ΄ < 1-(1-1/k)^{k-π}. A breakthrough by Keevash, Knox and Mycroft [STOC '13] resolved this problem for π = k-1 by showing that PM(k, k-1, Ξ΄) is in P for Ξ΄ > 1/k. Based on their result for π = k-1, Keevash, Knox and Mycroft conjectured that PM(k, π, Ξ΄) is in P for every Ξ΄ > 1-(1-1/k)^{k-π}.
In this paper it is shown that this decision problem for perfect matchings can be reduced to the study of the minimum π-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 π β₯ 0.4k.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol229-icalp2022/LIPIcs.ICALP.2022.64/LIPIcs.ICALP.2022.64.pdf
Computational Complexity
Perfect Matching
Hypergraph