Document Open Access Logo

The Decision Problem for Perfect Matchings in Dense Hypergraphs

Authors Luyining Gan , Jie Han

PDF
Thumbnail PDF

File

LIPIcs.ICALP.2022.64.pdf
  • Filesize: 0.82 MB
  • 16 pages

Document Identifiers

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

Cite As Get BibTex

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) https://doi.org/10.4230/LIPIcs.ICALP.2022.64

Abstract

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
Keywords
  • Computational Complexity
  • Perfect Matching
  • Hypergraph

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Noga Alon, Peter Frankl, Hao Huang, Vojtech Rödl, Andrzej Ruciński, and Benny Sudakov. Large matchings in uniform hypergraphs and the conjecture of Erdős and Samuels. J. Combin. Theory Ser. A, 119(6):1200-1215, 2012. URL: https://doi.org/10.1016/j.jcta.2012.02.004.
  2. Arash Asadpour, Uriel Feige, and Amin Saber. Santa claus meets hypergraph matchings. Approximation, randomization and combinatorial optimization, Lecture Notes in Comput. Sci., vol. 5171, Springer, Berlin, pages 10-20, 2008. Google Scholar
  3. Yulin Chang, Huifen Ge, Jie Han, and Guanghui Wang. Matching of given sizes in hypergraphs. ArXiv eprint: 2106.16068, 2021. Google Scholar
  4. Laihao Ding, Jie Han, Shumin Sun, Guanghui Wang, and Wenling Zhou. F-factors in quasirandom hypergraphs. J. London Math. Soc., to appear, 2022. Google Scholar
  5. Jack Edmonds. Paths, trees, and flowers. Canad. J. Math., 17:449-467, 1965. Google Scholar
  6. Peter Frankl and Andrey Kupavskii. The Erdős Matching Conjecture and concentration inequalities. ArXiv e-prints, 2018. URL: http://arxiv.org/abs/1806.08855.
  7. Frederik Garbe and Richard Mycroft. Hamilton cycles in hypergraphs below the Dirac threshold. J. Combin. Theory Ser. B, 133:153-210, 2018. URL: https://doi.org/10.1016/j.jctb.2018.04.010.
  8. Stefan Glock, Daniela Kühn, Allan Lo, and Deryk Osthus. The existence of designs via iterative absorption: hypergraph F-designs for arbitrary F. Mem. Amer. Math. Soc., accepted, 2020. Google Scholar
  9. Hiep Hàn, Yury Person, and Mathias Schacht. On perfect matchings in uniform hypergraphs with large minimum vertex degree. SIAM J. Discrete Math, 23:732-748, 2009. Google Scholar
  10. Jie Han. Perfect matchings in hypergraphs and the Erdős matching conjecture. SIAM J. Discrete Math., 30(3):1351-1357, 2016. URL: https://doi.org/10.1137/16M1056079.
  11. Jie Han. Decision problem for perfect matchings in dense k-uniform hypergraphs. Trans. Amer. Math. Soc., 369(7):5197-5218, 2017. Google Scholar
  12. Jie Han and Peter Keevash. Finding perfect matchings in dense hypergraphs. In Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2366-2377, 2020. URL: https://doi.org/10.1137/1.9781611975994.145.
  13. Jie Han and Andrew Treglown. The complexity of perfect matchings and packings in dense hypergraphs. J. Combin. Theory Ser. B, 141:72-104, 2020. URL: https://doi.org/10.1016/j.jctb.2019.06.004.
  14. Richard M. Karp. Reducibility among combinatorial problems. In Complexity of computer computations (Proc. Sympos., IBM Thomas J. Watson Res. Center, Yorktown Heights, N.Y., 1972), pages 85-103. Plenum, New York, 1972. Google Scholar
  15. Marek Karpiński, Andrzej Ruciński, and Edyta Szymańska. Computational complexity of the perfect matching problem in hypergraphs with subcritical density. Internat. J. Found. Comput. Sci., 21(6):905-924, 2010. URL: https://doi.org/10.1142/S0129054110007635.
  16. Peter Keevash. The existence of designs. preprint, 2014. Google Scholar
  17. Peter Keevash, Fiachra Knox, and Richard Mycroft. Polynomial-time perfect matchings in dense hypergraphs. Proceedings of the 45th STOC (2013), 2013. Google Scholar
  18. Peter Keevash, Fiachra Knox, and Richard Mycroft. Polynomial-time perfect matchings in dense hypergraphs. Adv. Math., 269:265-334, 2015. URL: https://doi.org/10.1016/j.aim.2014.10.009.
  19. Daniela Kühn, Deryk Osthus, and Timothy Townsend. Fractional and integer matchings in uniform hypergraphs. European J. Combin., 38:83-96, 2014. URL: https://doi.org/10.1016/j.ejc.2013.11.006.
  20. Allan Lo and Klas Markström. F-factors in hypergraphs via absorption. Graphs Combin., 31(3):679-712, 2015. URL: https://doi.org/10.1007/s00373-014-1410-8.
  21. Vojtech Rödl, Andrzej Ruciński, and Endre Szemerédi. An approximate Dirac-type theorem for k-uniform hypergraphs. Combinatorica, 28(2):229-260, 2008. Google Scholar
  22. Edyta Szymańska. The complexity of almost perfect matchings and other packing problems in uniform hypergraphs with high codegree. European J. Combin., 34(3):632-646, 2013. URL: https://doi.org/10.1016/j.ejc.2011.12.009.
  23. Raphael Yuster. Combinatorial and computational aspects of graph packing and graph decomposition. Computer Science Review, 1(1):12-26, 2007. URL: https://doi.org/10.1016/j.cosrev.2007.07.002.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact