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Finding a Maximum Restricted t-Matching via Boolean Edge-CSP

Authors Yuni Iwamasa , Yusuke Kobayashi , Kenjiro Takazawa

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ACM Subject Classification
  • Mathematics of computing → Combinatorial optimization
  • Theory of computation → Discrete optimization
Keywords
  • Polynomial algorithm
  • C_k-free 2-matching
  • Jump system
  • Boolean edge-CSP

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Abstract

The problem of finding a maximum 2-matching without short cycles has received significant attention due to its relevance to the Hamilton cycle problem. This problem is generalized to finding a maximum t-matching which excludes specified complete t-partite subgraphs, where t is a fixed positive integer. The polynomial solvability of this generalized problem remains an open question. In this paper, we present polynomial-time algorithms for the following two cases of this problem: in the first case the forbidden complete t-partite subgraphs are edge-disjoint; and in the second case the maximum degree of the input graph is at most 2t-1. Our result for the first case extends the previous work of Nam (1994) showing the polynomial solvability of the problem of finding a maximum 2-matching without cycles of length four, where the cycles of length four are vertex-disjoint. The second result expands upon the works of Bérczi and Végh (2010) and Kobayashi and Yin (2012), which focused on graphs with maximum degree at most t+1. Our algorithms are obtained from exploiting the discrete structure of restricted t-matchings and employing an algorithm for the Boolean edge-CSP.

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Yuni Iwamasa, Yusuke Kobayashi, and Kenjiro Takazawa. Finding a Maximum Restricted t-Matching via Boolean Edge-CSP. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 75:1-75:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ESA.2024.75

Author Details

Yuni Iwamasa
  • Kyoto University, Japan
Yusuke Kobayashi
  • Kyoto University, Japan
Kenjiro Takazawa
  • Hosei University, Japan

Funding

This work was partially supported by JSPS KAKENHI Grant Numbers JP20K11692, JP20K11699, JP22H05001, JP22K17854, JP24K02901, JP24K14828, and by JST ERATO Grant Number JPMJER2310.

Acknowledgements

The authors thank anonymous reviewers for their valuable comments.

References

  1. Kazutoshi Ando, Satoru Fujishige, and Takeshi Naitoh. A greedy algorithm for minimizing a separable convex function over a finite jump system. Journal of the Operations Research Society of Japan, 38(3):362-375, 1995. URL: https://doi.org/10.15807/jorsj.38.362.
  2. Maxim A. Babenko. Improved algorithms for even factors and square-free simple b-matchings. Algorithmica, 64:362-383, 2012. URL: https://doi.org/10.1007/s00453-012-9642-6.
  3. Kristóf Bérczi and Yusuke Kobayashi. An algorithm for (n-3)-connectivity augmentation problem: Jump system approach. Journal of Combinatorial Theory, Series B, 102:565-587, 2012. URL: https://doi.org/10.1016/j.jctb.2011.08.007.
  4. Kristóf Bérczi and László A. Végh. Restricted b-matchings in degree-bounded graphs. In F. Eisenbrand and B. Shepherd, editors, Integer Programming and Combinatorial Optimization: Proceedings of the 14th IPCO, LNCS 6080, pages 43-56. Springer, 2010. URL: https://doi.org/10.1007/978-3-642-13036-6_4.
  5. André Bouchet and William H. Cunningham. Delta-matroids, jump systems, and bisubmodular polyhedra. SIAM Journal on Discrete Mathematics, 8:17-32, 1995. URL: https://doi.org/10.1137/S0895480191222926.
  6. Gérard Cornuéjols and William Pulleyblank. A matching problem with side conditions. Discrete Mathematics, 29:135-159, 1980. URL: https://doi.org/10.1016/0012-365X(80)90002-3.
  7. András Frank. Restricted t-matchings in bipartite graphs. Discrete Applied Mathematics, 131:337-346, 2003. URL: https://doi.org/10.1016/S0166-218X(02)00461-4.
  8. David Hartvigsen. Extensions of Matching Theory. PhD thesis, Carnegie Mellon University, 1984. Google Scholar
  9. David Hartvigsen. Finding maximum square-free 2-matchings in bipartite graphs. Journal of Combinatorial Theory, Series B, 96:693-705, 2006. URL: https://doi.org/10.1016/j.jctb.2006.01.004.
  10. David Hartvigsen. Finding triangle-free 2-factors in general graphs. Journal of Graph Theory, available online, 2024. URL: https://doi.org/10.1002/jgt.23089.
  11. David Hartvigsen and Yanjun Li. Maximum cardinality simple 2-matchings in subcubic graphs. SIAM Journal on Optimization, 21:1027-1045, 2011. URL: https://doi.org/10.1137/090760416.
  12. Alexandr Kazda, Vladimir Kolmogorov, and Michal Rolínek. Even delta-matroids and the complexity of planar boolean CSPs. ACM Transactions on Algorithms, 15(2):22:1-22:33, 2019. URL: https://doi.org/10.1145/3230649.
  13. Zoltán Király. C₄-free 2-factors in bipartite graphs. Technical Report TR-2001-13, Egerváry Research Group, 1999. www.cs.elte.hu/egres. Google Scholar
  14. Zoltán Király. Restricted t-matchings in bipartite graphs. Technical Report QP-2009-04, Egerváry Research Group, 2009. www.cs.elte.hu/egres. Google Scholar
  15. Yusuke Kobayashi, Kazuo Murota, and Ken'ichiro Tanaka. Operations on M-convex functions on jump systems. SIAM Journal on Discrete Mathematics, 21(1):107-129, 2007. URL: https://doi.org/10.1137/060652841.
  16. Yusuke Kobayashi and Takashi Noguchi. An approximation algorithm for two-edge-connected subgraph problem via triangle-free two-edge-cover. In Satoru Iwata and Naonori Kakimura, editors, 34th International Symposium on Algorithms and Computation, ISAAC 2023, December 3-6, 2023, Kyoto, Japan, volume 283 of LIPIcs, pages 49:1-49:10. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2023. URL: https://doi.org/10.4230/LIPICS.ISAAC.2023.49.
  17. Yusuke Kobayashi, Jaćint Szabó, and Kenjiro Takazawa. A proof of Cunningham’s conjecture on restricted subgraphs and jump systems. Journal of Combinatorial Theory, Series B, 102:948-966, 2012. URL: https://doi.org/10.1016/J.JCTB.2012.03.003.
  18. Yusuke Kobayashi and Xin Yin. An algorithm for finding a maximum t-matching excluding complete partite subgraphs. Discrete Optimization, 9:98-108, 2012. URL: https://doi.org/10.1016/J.DISOPT.2012.02.003.
  19. László Lovász. The membership problem in jump systems. Journal of Combinatorial Theory, Series B, 70(1):45-66, 1997. URL: https://doi.org/10.1006/jctb.1997.1744.
  20. Kazuo Murota. M-convex functions on jump systems: A general framework for minsquare graph factor problem. SIAM Journal on Discrete Mathematics, 20(1):213-226, 2006. URL: https://doi.org/10.1137/040618710.
  21. Kazuo Murota. On basic operations related to network induction of discrete convex functions. Optimization Methods and Software, 36(2-3):519-559, 2021. URL: https://doi.org/10.1080/10556788.2020.1818080.
  22. Yunsun Nam. Matching Theory: Subgraphs with Degree Constraints and Other Properties. PhD thesis, University of British Columbia, 1994. Google Scholar
  23. Katarzyna Paluch and Mateusz Wasylkiewicz. A simple combinatorial algorithm for restricted 2-matchings in subcubic graphs - via half-edges. Information Processing Letters, 171:106146, 2021. URL: https://doi.org/10.1016/j.ipl.2021.106146.
  24. Gyula Pap. Combinatorial algorithms for matchings, even factors and square-free 2-factors. Mathematical Programming, 110:57-69, 2007. URL: https://doi.org/10.1007/s10107-006-0053-9.
  25. Francesca Rossi, Peter van Beek, and Toby Walsh, editors. Handbook of Constraint Programming, volume 2 of Foundations of Artificial Intelligence. Elsevier, 2006. Google Scholar
  26. Akiyoshi Shioura and Ken'ichiro Tanaka. Polynomial-time algorithms for linear and convex optimization on jump systems. SIAM Journal on Discrete Mathematics, 21(2):504-522, 2007. URL: https://doi.org/10.1137/060656899.
  27. Kenjiro Takazawa. A weighted K_t,t-free t-factor algorithm for bipartite graphs. Mathematics of Operations Research, 34:351-362, 2009. URL: https://doi.org/doi.org/10.1287/moor.1080.0365.
  28. Kenjiro Takazawa. Decomposition theorems for square-free 2-matchings in bipartite graphs. Discrete Applied Mathematics, 233:215-223, 2017. URL: https://doi.org/10.1016/j.dam.2017.07.035.
  29. Kenjiro Takazawa. Finding a maximum 2-matching excluding prescribed cycles in bipartite graphs. Discrete Optimization, 26:26-40, 2017. URL: https://doi.org/10.1016/j.disopt.2017.05.003.
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