Restricted t-Matchings via Half-Edges

Authors Katarzyna Paluch, Mateusz Wasylkiewicz

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Katarzyna Paluch
  • Institute of Computer Science, University of Wrocław, Poland
Mateusz Wasylkiewicz
  • Institute of Computer Science, University of Wrocław, Poland


The authors thank Pratik Ghosal and Prajakta Nimbhorkar for discussions at the early stage of research on this paper.

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Katarzyna Paluch and Mateusz Wasylkiewicz. Restricted t-Matchings via Half-Edges. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 73:1-73:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


For a bipartite graph G we consider the problem of finding a maximum size/weight square-free 2-matching and its generalization - the problem of finding a maximum size/weight K_{t,t}-free t-matching, where t is an integer greater than two and K_{t,t} denotes a bipartite clique with t vertices on each of the two sides. Since the weighted versions of these problems are NP-hard in general, we assume that the weights are vertex-induced on any subgraph isomorphic to K_{t,t}. We present simple combinatorial algorithms for these problems. Our algorithms are significantly simpler and faster than those previously known. We dispense with the need to shrink squares and, more generally subgraphs isomorphic to K_{t,t}, the operation which occurred in all previous algorithms for such t-matchings and instead use so-called half-edges. A half-edge of edge e is, informally speaking, a half of e containing exactly one of its endpoints. Additionally, we consider another problem concerning restricted matchings. Given a (not necessarily bipartite) graph G = (V,E), a set of k subsets of edges E₁, E₂, …, E_k and k natural numbers r₁, r₂, …, r_k, the Restricted Matching Problem asks to find a maximum size matching of G among such ones that for each 1 ≤ i ≤ k, M contains at most r_i edges of E_i. This problem is NP-hard even when G is bipartite. We show that it is solvable in polynomial time if (i) for each i the graph G contains a clique or a bipartite clique on all endpoints of E_i; in the case of a bipartite clique it is required to contain E_i and (ii) the sets E₁, …, E_k are almost vertex-disjoint - the endpoints of any two different sets have at most one vertex in common.

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ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • restricted 2-matchings


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  1. Maxim Babenko. Improved algorithms for even factors and square-free simple b-matchings. Algorithmica, 64(3):362-383, 2012. Google Scholar
  2. Kristóf Bérczi and Yusuke Kobayashi. An algorithm for (n-3)-connectivity augmentation problem: Jump system approach. Journal of Combinatorial Theory, Series B, 102(3):565-587, 2012. Google Scholar
  3. Kristóf Bérczi and László Végh. Restricted b-matchings in degree-bounded graphs. In Integer Programming and Combinatorial Optimization, pages 43-56, 2010. Google Scholar
  4. Gérard Cornuéjols and William Pulleyblank. A matching problem with side conditions. Discrete Mathematics, 29(2):135-159, 1980. Google Scholar
  5. Marshall Fisher, George Nemhauser, and Laurence Wolsey. An analysis of approximations for finding a maximum weight hamiltonian circuit. Operations Research, 27(4):799-809, 1979. Google Scholar
  6. Tamás Fleiner. Uncrossing a family of set-pairs. Combinatorica, 21:145-150, 2001. Google Scholar
  7. András Frank. Restricted t-matchings in bipartite graphs. Discrete Applied Mathematics, 131(2):337-346, 2003. Google Scholar
  8. András Frank and Tibor Jordán. Minimal edge-coverings of pairs of sets. Journal of Combinatorial Theory, Series B, 65(1):73-110, 1995. Google Scholar
  9. Michael Fredman and Robert Endre Tarjan. Fibonacci heaps and their uses in improved network optimization algorithms. Journal of the ACM, 34(3):596-615, 1987. Google Scholar
  10. Zvi Galil and Giuseppe Italiano. Reducing edge connectivity to vertex connectivity. SIGACT News, 22:57-61, 1991. Google Scholar
  11. Michael Garey and David Johnson. Computers and Intractability. Freeman, 1979. Google Scholar
  12. Jim Geelen. The C₆-free 2-factor problem in bipartite graphs is NP-complete. Unpublished, 1999. Google Scholar
  13. David Hartvigsen. Extensions of Matching Theory. PhD thesis, Carnegie-Mellon University, 1984. Google Scholar
  14. David Hartvigsen. Finding maximum square-free 2-matchings in bipartite graphs. Journal of Combinatorial Theory, Series B, 96(5):693-705, 2006. Google Scholar
  15. David Hartvigsen and Yanjun Li. Maximum cardinality simple 2-matchings in subcubic graphs. SIAM Journal on Optimization, 21(3):1027-1045, 2011. Google Scholar
  16. David Hartvigsen and Yanjun Li. Polyhedron of triangle-free simple 2-matchings in subcubic graphs. Mathematical Programming, 138:43-82, 2013. Google Scholar
  17. Alon Itai, Michael Rodeh, and Steven Tanimoto. Some matching problems for bipartite graphs. Journal of the ACM, 25(4):517-525, 1978. Google Scholar
  18. Zoltán Király. C₄-free 2-factors in bipartite graphs. Technical report, Egerváry Research Group, 1999. Google Scholar
  19. Zoltán Király. Restricted t-matchings in bipartite graphs. Technical report, Egerváry Research Group, 2009. Google Scholar
  20. Yusuke Kobayashi. A simple algorithm for finding a maximum triangle-free 2-matching in subcubic graphs. Discrete Optimization, 7:197-202, 2010. Google Scholar
  21. Yusuke Kobayashi. Weighted triangle-free 2-matching problem with edge-disjoint forbidden triangles. In Integer Programming and Combinatorial Optimization, pages 280-293, 2020. Google Scholar
  22. Yusuke Kobayashi and Xin Yin. An algorithm for finding a maximum t-matching excluding complete partite subgraphs. Discrete Optimization, 9(2):98-108, 2012. Google Scholar
  23. László Lovász and Michael Plummer. Matching theory. AMS Chelsea Publishing, corrected reprint of the 1986 original edition, 2009. Google Scholar
  24. Márton Makai. On maximum cost K_t,t-free t-matchings of bipartite graphs. SIAM Journal on Discrete Mathematics, 21:349-360, 2007. Google Scholar
  25. Monaldo Mastrolilli and Georgios Stamoulis. Constrained matching problems in bipartite graphs. In Combinatorial Optimization, pages 344-355, 2012. Google Scholar
  26. Yunsun Nam. Matching Theory: Subgraphs with Degree Constraints and other Properties. PhD thesis, University of British Columbia, 1994. Google Scholar
  27. Katarzyna Paluch, Khaled Elbassioni, and Anke van Zuylen. Simpler approximation of the maximum asymmetric traveling salesman problem. In 29th International Symposium on Theoretical Aspects of Computer Science, pages 501-506, 2012. Google Scholar
  28. Katarzyna Paluch and Mateusz Wasylkiewicz. A simple combinatorial algorithm for restricted 2-matchings in subcubic graphs - via half-edges. Information Processing Letters, 171, 2021. Google Scholar
  29. Gyula Pap. Alternating paths revisited ii: restricted b-matchings in bipartite graphs. Technical report, Egerváry Research Group, 2005. Google Scholar
  30. Gyula Pap. Combinatorial algorithms for matchings, even factors and square-free 2-factors. Mathematical Programming, 110:57-69, 2007. Google Scholar
  31. Irena Rusu. Maximum weight edge-constrained matchings. Discrete Applied Mathematics, 156(5):662-672, 2008. Google Scholar
  32. Alexander Schrijver. Combinatorial optimization: polyhedra and efficiency, volume 24. Springer Science & Business Media, 2003. Google Scholar
  33. Kenjiro Takazawa. A weighted K_t,t-free t-factor algorithm for bipartite graphs. Mathematics of Operations Research, 34(2):351-362, 2009. Google Scholar
  34. Kenjiro Takazawa. Decomposition theorems for square-free 2-matchings in bipartite graphs. Discrete Applied Mathematics, 233:215-223, 2017. Google Scholar
  35. Kenjiro Takazawa. Finding a maximum 2-matching excluding prescribed cycles in bipartite graphs. Discrete Optimization, 26:26-40, 2017. Google Scholar
  36. László Végh and András Benczúr. Primal-dual approach for directed vertex connectivity augmentation and generalizations. In SODA 2005. Proceedings of the sixteenth annual ACM-SIAM symposium on discrete algorithms., pages 186-194, 2005. Google Scholar
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