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The Parameterized Complexity of the Induced Matching Problem in Planar Graphs

  • Hannes Moser
  • Somnath Sikdar
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4613)

Abstract

Given a graph G and an integer k ≥ 0, the NP-complete Induced Matching problem asks for an edge subset M such that M is a matching and no two edges of M are joined by an edge of G. The complexity of this problem on general graphs as well as on many restricted graph classes has been studied intensively. However, little is known about the parameterized complexity of this problem. Our main contribution is to show that Induced Matching, which is W[1]-hard in general, admits a linear problem kernel on planar graphs. Additionally, we generalize a known algorithm for Induced Matching on trees to graphs of bounded treewidth using an improved dynamic programming approach.

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References

  1. 1.
    Alber, J., Fan, H., Fellows, M.R., Fernau, H., Niedermeier, R., Rosamond, F.A., Stege, U.: A refined search tree technique for dominating set on planar graphs. Journal of Computer and System Sciences 71(4), 385–405 (2005)zbMATHCrossRefGoogle Scholar
  2. 2.
    Alber, J., Fellows, M.R., Niedermeier, R.: Polynomial-time data reduction for dominating set. Journal of the ACM 51(3), 363–384 (2004)CrossRefGoogle Scholar
  3. 3.
    Alber, J., Niedermeier, R.: Improved tree decomposition based algorithms for domination-like problems. In: Rajsbaum, S. (ed.) LATIN 2002. LNCS, vol. 2286, pp. 613–628. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  4. 4.
    Bodlaender, H.L.: Treewidth: Characterizations, applications, and computations. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 1–14. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. 5.
    Cameron, K.: Induced matchings in intersection graphs. Discrete Mathematics 278(1-3), 1–9 (2004)zbMATHCrossRefGoogle Scholar
  6. 6.
    Cameron, K., Sritharan, R., Tang, Y.: Finding a maximum induced matching in weakly chordal graphs. Discrete Mathematics 266(1-3), 133–142 (2003)zbMATHCrossRefGoogle Scholar
  7. 7.
    Chen, J., Fernau, H., Kanj, I.A., Xia, G.: Parametric duality and kernelization: lower bounds and upper bounds on kernel size. SIAM Journal on Computing (to appear)Google Scholar
  8. 8.
    Chlebík, M., Chlebíková, J.: Approximation hardness of dominating set problems. In: Albers, S., Radzik, T. (eds.) ESA 2004. LNCS, vol. 3221, pp. 192–203. Springer, Heidelberg (2004)Google Scholar
  9. 9.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)Google Scholar
  10. 10.
    Duckworth, W., Manlove, D., Zito, M.: On the approximability of the maximum induced matching problem. Journal of Discrete Algorithms 3(1), 79–91 (2005)zbMATHCrossRefGoogle Scholar
  11. 11.
    Fomin, F.V., Thilikos, D.M.: Fast parameterized algorithms for graphs on surfaces: Linear kernel and exponential speed-up. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 581–592. Springer, Heidelberg (2004)Google Scholar
  12. 12.
    Golumbic, M.C., Lewenstein, M.: New results on induced matchings. Discrete Applied Mathematics 101(1-3), 157–165 (2000)zbMATHCrossRefGoogle Scholar
  13. 13.
    Gotthilf, Z., Lewenstein, M.: Tighter approximations for maximum induced matchings in regular graphs. In: Erlebach, T., Persinao, G. (eds.) WAOA 2005. LNCS, vol. 3879, pp. 270–281. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  14. 14.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. SIGACT News 38(1), 31–45 (2007)CrossRefGoogle Scholar
  15. 15.
    Guo, J., Niedermeier, R.: Linear problem kernels for NP-hard problems on planar graphs. In: Arge, L., Cachin, C., Jurdzinski, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 375–386. Springer, Heidelberg (2007)Google Scholar
  16. 16.
    Guo, J., Niedermeier, R., Wernicke, S.: Fixed-parameter tractability results for full-degree spanning tree and its dual. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 203–214. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  17. 17.
    Ko, C.W., Shepherd, F.B.: Bipartite domination and simultaneous matroid covers. SIAM Journal on Discrete Mathematics 16(4), 517–523 (2003)zbMATHCrossRefGoogle Scholar
  18. 18.
    Kobler, D., Rotics, U.: Finding maximum induced matchings in subclasses of claw-free and P 5-free graphs, and in graphs with matching and induced matching of equal maximum size. Algorithmica 37(4), 327–346 (2003)zbMATHCrossRefGoogle Scholar
  19. 19.
    Lozin, V.V., Rautenbach, D.: Some results on graphs without long induced paths. Information Processing Letters 88(4), 167–171 (2003)CrossRefGoogle Scholar
  20. 20.
    Moser, H., Thilikos, D.M.: Parameterized complexity of finding regular induced subgraphs. In: Proceedings of ACiD 2006. Texts in Algorithmics, vol. 7, pp. 107–118. College Publications (2006)Google Scholar
  21. 21.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)zbMATHGoogle Scholar
  22. 22.
    Stockmeyer, L.J., Vazirani, V.V.: NP-completeness of some generalizations of the maximum matching problem. Information Processing Letters 15(1), 14–19 (1982)zbMATHCrossRefGoogle Scholar
  23. 23.
    Zito, M.: Induced matchings in regular graphs and trees. In: Widmayer, P., Neyer, G., Eidenbenz, S. (eds.) WG 1999. LNCS, vol. 1665, pp. 89–100. Springer, Heidelberg (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Hannes Moser
    • 1
  • Somnath Sikdar
    • 2
  1. 1.Institut für Informatik, Friedrich-Schiller-Universität Jena, Ernst-Abbe-Platz 2, D-07743 JenaGermany
  2. 2.The Institute of Mathematical Sciences, C.I.T Campus, Taramani, Chennai 600113India

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