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Quadratic Vertex Kernel for Rainbow Matching

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Abstract

In this paper, we study the NP-complete colorful variant of the classic matching problem, namely, the Rainbow Matching problem. Given an edge-colored graph G and a positive integer k, the goal is to decide whether there exists a matching of size at least k such that the edges in the matching have distinct colors. Previously, in [MFCS’17], we studied this problem from the view point of Parameterized Complexity and gave efficient FPT algorithms as well as a quadratic kernel on paths. In this paper we design a quadratic vertex kernel for Rainbow Matching on general graphs; generalizing the earlier quadratic kernel on paths to general graphs. For our kernelization algorithm we combine a graph decomposition method with an application of expansion lemma.

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Notes

  1. (I, k) is a Yes-instance if and only if \((I',k')\) is a Yes–instance.

References

  1. Aharoni, R., Berger, E.: Rainbow matchings in \( r \)-partite \( r \)-graphs. Electron. J. Combin. 16(1), 119 (2009)

    Article  MathSciNet  Google Scholar 

  2. Alon, N.: Multicolored matchings in hypergraphs. Mosc. J. Combin. Number Theory 1, 3–10 (2011)

    MathSciNet  MATH  Google Scholar 

  3. Cygan, M., Fomin, F.V., Kowalik, L., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Berlin (2015)

    Book  Google Scholar 

  4. Diemunsch, J., Ferrara, M., Moffatt, C., Pfender, F., Wenger, P.S.: Rainbow matchings of size\(\backslash \)delta (g) in properly edge-colored graphs. arXiv preprint arXiv:1108.2521 (2011)

  5. Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Springer, Berlin (2013)

    Book  Google Scholar 

  6. Edmonds, J.: Paths, trees, and flowers. Can. J. Math. 17(3), 449–467 (1965)

    Article  MathSciNet  Google Scholar 

  7. Fomin, F.V., Lokshtanov, D., Saurabh, S., Zehavi, M.: Kernelization: Theory of Parameterized Preprocessing. Cambridge University Press, Cambridge (2018)

    Book  Google Scholar 

  8. Fujita, S., Kaneko, A., Schiermeyer, I., Suzuki, K.: A rainbow \( k \)-matching in the complete graph with \( r \) colors. Electron. J. Combin. 16(1), 51 (2009)

    Article  MathSciNet  Google Scholar 

  9. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, San Francisco (1979)

    MATH  Google Scholar 

  10. Glebov, R., Sudakov, B., Szabó, T.: How many colors guarantee a rainbow matching? arXiv preprint arXiv:1211.0793 (2012)

  11. Gupta, S., Roy, S., Saurabh, S., Zehavi, M.: Parameterized algorithms and kernels for rainbow matching. Algorithmica 81, 1684–1698 (2019). https://doi.org/10.1007/s00453-018-0497-3

    Article  MathSciNet  MATH  Google Scholar 

  12. Itai, A., Rodeh, M., Tanimoto, S.L.: Some matching problems for bipartite graphs. J. ACM 25(4), 517–525 (1978)

    Article  MathSciNet  Google Scholar 

  13. Kano, M., Li, X.: Monochromatic and heterochromatic subgraphs in edge-colored graphs-a survey. Graphs Combin. 24(4), 237–263 (2008)

    Article  MathSciNet  Google Scholar 

  14. Kostochka, A., Yancey, M.: Large rainbow matchings in edge-coloured graphs. Combin. Probab. Comput. 21(1–2), 255–263 (2012)

    Article  MathSciNet  Google Scholar 

  15. Le, V.B., Pfender, F.: Complexity results for rainbow matchings. Theor. Comput. Sci. 524, 27–33 (2014)

    Article  MathSciNet  Google Scholar 

  16. LeSaulnier, T.D., Stocker, C., Wenger, P.S., West, D.B.: Rainbow matching in edge-colored graphs. Electron. J. Combin. 17(1), 26 (2010)

    MathSciNet  MATH  Google Scholar 

  17. Lo, A.: Existences of rainbow matchings and rainbow matching covers. Discrete Math. 338(11), 2119–2124 (2015)

    Article  MathSciNet  Google Scholar 

  18. Lovász, L., Plummer, M.D.: Matching Theory, vol. 367. American Mathematical Society, Providence (2009)

    MATH  Google Scholar 

  19. Micali, S., Vazirani, V.V.: An \(O(\sqrt{|V|} |E|)\) algorithm for finding maximum matching in general graphs. In: 21st Annual Symposium on Foundations of Computer Science, Syracuse, NY, USA, 13–15 Oct 1980, pp. 17–27 (1980)

  20. Pokrovskiy, A.: An approximate version of a conjecture of Aharoni and Berger. Adv. Math. 333, 1197–1241 (2018)

    Article  MathSciNet  Google Scholar 

  21. Ryser, H.J.: Neuere probleme der kombinatorik. Vorträge über Kombinatorik, Oberwolfach, pp. 69–91 (1967)

  22. Stockmeyer, L.J., Vazirani, V.V.: Np-completeness of some generalizations of the maximum matching problem. Inf. Process. Lett. 15(1), 14–19 (1982)

    Article  MathSciNet  Google Scholar 

  23. Wang, G.: Rainbow matchings in properly edge colored graphs. Electron. J. Combin. 18(1), 162 (2011)

    MathSciNet  MATH  Google Scholar 

  24. Wang, G., Li, H.: Heterochromatic matchings in edge-colored graphs. Electron. J. Combin. 15(1), 138 (2008)

    Article  MathSciNet  Google Scholar 

  25. Yannakakis, M., Gavril, F.: Edge dominating sets in graphs. SIAM J. Appl. Math. 38(3), 364–372 (1980)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. National Institute for Science Education and Research, Bhubaneswar, India

    Sushmita Gupta

  2. The Institute of Mathematical Sciences, HBNI, Chennai, India

    Sanjukta Roy & Saket Saurabh

  3. University of Bergen, Bergen, Norway

    Saket Saurabh

  4. Ben-Gurion University, Beersheba, Israel

    Meirav Zehavi

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  1. Sushmita Gupta
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  2. Sanjukta Roy
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  3. Saket Saurabh
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  4. Meirav Zehavi
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Gupta, S., Roy, S., Saurabh, S. et al. Quadratic Vertex Kernel for Rainbow Matching. Algorithmica 82, 881–897 (2020). https://doi.org/10.1007/s00453-019-00618-0

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