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Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2022.64
URN: urn:nbn:de:0030-drops-164057
URL: https://drops.dagstuhl.de/opus/volltexte/2022/16405/
Gan, Luyining ;
Han, Jie
The Decision Problem for Perfect Matchings in Dense Hypergraphs
Abstract
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.
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: |
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Computational Complexity, Perfect Matching, Hypergraph |
Collection: |
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49th International Colloquium on Automata, Languages, and Programming (ICALP 2022) |
Issue Date: |
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2022 |
Date of publication: |
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28.06.2022 |