Linear Time Algorithm for Computing a Small Biclique in Graphs without Long Induced Paths

  • Aistis Atminas
  • Vadim V. Lozin
  • Igor Razgon
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7357)

Abstract

The biclique problem asks, given a graph G and a parameter k, whether G has a complete bipartite subgraph of k vertices in each part (a biclique of order k). Fixed-parameter tractability of this problem is a longstanding open question in parameterized complexity that received a lot of attention from the community. In this paper we consider a restricted version of this problem by introducing an additional parameter s and assuming that G does not have induced (i.e. chordless) paths of length s. We prove that under this parameterization the problem becomes fixed-parameter linear. The main tool in our proof is a Ramsey-type theorem stating that a graph with a long (not necessarily induced) path contains either a long induced path or a large biclique.

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References

  1. 1.
    Arbib, C., Mosca, R.: Polynomial algorithms for special cases of the balanced complete bipartite subgraph problem. J. Combin. Math. Combin. Comput. 39, 3–22 (1999)MathSciNetGoogle Scholar
  2. 2.
    Bascó, G., Tuza, Z.: A characterization of graphs without long induced paths. J. of Graph Theory 14, 455–464 (1990)MATHCrossRefGoogle Scholar
  3. 3.
    Binkele-Raible, D., Fernau, H., Gaspers, S., Liedloff, M.: Exact exponential-time algorithms for finding bicliques. Inf. Process. Lett. 111(2), 64–67 (2010)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bodlaender, H.L.: On linear time minor tests with depth-first search. J. Algorithms 14(1), 1–23 (1993)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Broersma, H., Golovach, P.A., Paulusma, D., Song, J.: Updating the complexity status of coloring graphs without a fixed induced linear forest. Theor. Comput. Sci. 414(1), 9–19 (2012)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Bulatov, A.A., Marx, D.: Constraint Satisfaction Parameterized by Solution Size. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011, Part I. LNCS, vol. 6755, pp. 424–436. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  7. 7.
    Chen, Y., Thurley, M., Weyer, M.: Understanding the Complexity of Induced Subgraph Isomorphisms. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 587–596. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  8. 8.
    Courcelle, B., Makowsky, J.A., Rotics, U.: Linear Time Solvable Optimization Problems on Graphs of Bounded Clique Width. In: Hromkovič, J., Sýkora, O. (eds.) WG 1998. LNCS, vol. 1517, pp. 1–16. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  9. 9.
    Demaine, E., Gutin, G.Z., Marx, D., Stege, U.: 07281 open problems – structure theory and FPT algorithmcs for graphs, digraphs and hypergraphs. In: Demaine, E., Gutin, G.Z., Marx, D., Stege, U. (eds.) Structure Theory and FPT Algorithmics for Graphs, Digraphs and Hypergraphs, Dagstuhl, Germany. Dagstuhl Seminar Proceedings, vol. (07281), Internationales Begegnungs- und Forschungszentrum für Informatik (IBFI), Schloss Dagstuhl, Germany (2007)Google Scholar
  10. 10.
    Diestel, R.: Graph Theory, 3rd edn. Springer (2005)Google Scholar
  11. 11.
    Dong, J.: Some results on graphs without long induced paths. J. of Graph Theory 22, 23–28 (1996)MATHCrossRefGoogle Scholar
  12. 12.
    Feige, U., Kogan, S.: Hardness of approximation of the balanced complete bipartite subgraph problem. Technical Report MCS04-04, Weizmann Institute of Science (2004)Google Scholar
  13. 13.
    Fellows, M., Gaspers, S., Rosamond, F.: Multivariate complexity theory. In: Blum, E.K., Aho, A.V. (eds.) Computer Science: The Hardware, Software and Heart of It, pp. 269–293. Springer (2011)Google Scholar
  14. 14.
    Fellows, M.R., Langston, M.A.: On search, decision and the efficiency of polynomial-time algorithms (extended abstract). In: STOC, pp. 501–512 (1989)Google Scholar
  15. 15.
    Golovach, P.A., Paulusma, D., Song, J.: Coloring Graphs without Short Cycles and Long Induced Paths. In: Owe, O., Steffen, M., Telle, J.A. (eds.) FCT 2011. LNCS, vol. 6914, pp. 193–204. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  16. 16.
    Johnson, D.S.: The NP-completeness column: An ongoing guide. J. Algorithms 8(3), 438–448 (1987)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Korobitsyn, D.: On the complexity of determining the domination number in monogenic classes of graphs. Diskretnaya Matematika 2(3), 90–96 (1990) (in Russian)MATHGoogle Scholar
  18. 18.
    Král’, D., Kratochvíl, J., Tuza, Z., Woeginger, G.J.: Complexity of Coloring Graphs without Forbidden Induced Subgraphs. In: Brandstädt, A., Le, V.B. (eds.) WG 2001. LNCS, vol. 2204, pp. 254–262. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  19. 19.
    Kühn, D., Osthus, D.: Induced subdivisions in K\(_{\mbox{s, s}}\)-free graphs of large average degree. Combinatorica 24(2), 287–304 (2004)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Le, V.B., Randerath, B., Schiermeyer, I.: On the complexity of 4-coloring graphs without long induced paths. Theor. Comput. Sci. 389(1-2), 330–335 (2007)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Lozin, V.V., Mosca, R.: Maximum independent sets in subclasses of P\(_{\mbox{5}}\)-free graphs. Inf. Process. Lett. 109(6), 319–324 (2009)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Lozin, V.V., Rautenbach, D.: Some results on graphs without long induced paths. Inf. Process. Lett. 88(4), 167–171 (2003)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Murphy, O.J.: Computing independent sets in graphs with large girth. Discrete Applied Mathematics 35(2), 167–170 (1992)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Randerath, B., Schiermeyer, I.: 3-colorability in P for P\(_{\mbox{6}}\)-free graphs. Discrete Applied Mathematics 136(2-3), 299–313 (2004)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Woeginger, G.J., Sgall, J.: The complexity of coloring graphs without long induced paths. Acta Cybernetica 15(1), 107–117 (2001)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Aistis Atminas
    • 1
  • Vadim V. Lozin
    • 1
  • Igor Razgon
    • 2
  1. 1.DIMAP and Mathematics InstituteUniversity of WarwickCoventryUK
  2. 2.Department of Computer ScienceUniversity of LeicesterLeicesterUK

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