Covering Graphs with Few Complete Bipartite Subgraphs

  • Herbert Fleischner
  • Egbert Mujuni
  • Daniel Paulusma
  • Stefan Szeider
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4855)

Abstract

Given a graph and an integer k, the biclique cover problem asks whether the edge-set of the given graph can be covered with at most k bicliques (complete bipartite subgraphs); the biclique vertex-cover problem asks whether the vertex-set of the given graph can be covered with at most k bicliques. Both problems are known to be NP-complete even if the given graph is bipartite. In this paper we investigate these two problems in the framework of parameterized complexity: do the problems become easier if k is assumed to be small? We show that, considering k as the parameter, the first problem is fixed-parameter tractable, while the second one is not fixed-parameter tractable unless P=NP.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Amilhastre, J., Vilarem, M.C., Janssen, P.: Complexity of minimum biclique cover and minimum biclique decomposition for bipartite domino-free graphs. Discr. Appl. Math. 86(2-3), 125–144 (1998)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Chartrand, G., Lesniak, L.: Graphs & digraphs, 4th edn. Chapman & Hall/CRC, Boca Raton, FL (2005)MATHGoogle Scholar
  3. 3.
    Cornaz, D., Fonlupt, J.: Chromatic characterization of biclique covers. Discrete Math. 306(5), 495–507 (2006)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Dantas, S., de Figueiredo, C.M., Gravier, S., Klein, S.: Finding H-partitions efficiently. RAIRO - Theoretical Informatics and Applications 39(1), 133–144 (2005)MATHCrossRefGoogle Scholar
  5. 5.
    Diestel, R.: Graph Theory, 2nd edn. Graduate Texts in Mathematics, vol. 173. Springer, New York (2000)Google Scholar
  6. 6.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. In: Monographs in Computer Science, Springer, Heidelberg (1999)Google Scholar
  7. 7.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. In: Texts in Theoretical Computer Science. An EATCS Series, vol. XIV, Springer, Heidelberg (2006)Google Scholar
  8. 8.
    Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Data reduction, exact, and heuristic algorithms for clique cover. In: Proc. ALENEX 2006, SIAM, pp. 86–94 (2006)Google Scholar
  9. 9.
    Gravier, S., Kobler, D., Kubiak, W.: Complexity of list coloring problems with a fixed total number of colors. Discr. Appl. Math. 117(1-3), 65–79 (2002)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Guo, J., Niedermeier, R.: Invitation to data reduction and problem kernelization. ACM SIGACT News 38(2), 31–45 (2007)CrossRefGoogle Scholar
  11. 11.
    Heydari, M.H., Morales, L., Shields Jr., C.O., Sudborough, I.H.: Computing cross associations for attack graphs and other applications. In: HICSS-40 2007. 40th Hawaii International International Conference on Systems Science, Waikoloa, Big Island, HI, USA, January 3-6, 2007, p. 270 (2007)Google Scholar
  12. 12.
    Hüffner, F., Niedermeier, R., Wernicke, S.: Techniques for practical fixed-parameter algorithms. The Computer Journal (in press, 2007) doi:10.1093/comjnl/bxm040Google Scholar
  13. 13.
    Müller, H.: On edge perfectness and classes of bipartite graphs. Discrete Math. 149(1-3), 159–187 (1996)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford (2006)Google Scholar
  15. 15.
    Orlin, J.: Contentment in graph theory: covering graphs with cliques. Nederl. Akad. Wetensch. Proc. Ser. A 80, Indag. Math. 39(5), 406–424 (1977)MathSciNetGoogle Scholar
  16. 16.
    Vikas, N.: Computational complexity of compaction to reflexive cycles. SIAM J. Comput. 32(1), 253–280 (2002/03)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Herbert Fleischner
    • 1
  • Egbert Mujuni
    • 2
  • Daniel Paulusma
    • 3
  • Stefan Szeider
    • 3
  1. 1.Department of Computer Science, Vienna Technical University, A-1040 ViennaAustria
  2. 2.Mathematics Department, University of Dar es Salaam, PO Box 35062, Dar es SalaamTanzania
  3. 3.Department of Computer Science, Durham University, Durham DH1 3LEUnited Kingdom

Personalised recommendations