Connected Coloring Completion for General Graphs: Algorithms and Complexity

  • Benny Chor
  • Michael Fellows
  • Mark A. Ragan
  • Igor Razgon
  • Frances Rosamond
  • Sagi Snir
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4598)

Abstract

An r-component connected coloring of a graph is a coloring of the vertices so that each color class induces a subgraph having at most r connected components. The concept has been well-studied for r = 1, in the case of trees, under the rubric of convex coloring, used in modeling perfect phylogenies. Several applications in bioinformatics of connected coloring problems on general graphs are discussed, including analysis of protein-protein interaction networks and protein structure graphs, and of phylogenetic relationships modeled by splits trees. We investigate the r-Component Connected Coloring Completion (r-CCC) problem, that takes as input a partially colored graph, having k uncolored vertices, and asks whether the partial coloring can be completed to an r-component connected coloring. For r = 1 this problem is shown to be NP-hard, but fixed-parameter tractable when parameterized by the number of uncolored vertices, solvable in time O*(8k). We also show that the 1-CCC problem, parameterized (only) by the treewidth t of the graph, is fixed-parameter tractable; we show this by a method that is of independent interest. The r-CCC problem is shown to be W[1]-hard, when parameterized by the treewidth bound t, for any r ≥ 2. Our proof also shows that the problem is NP-complete for r = 2, for general graphs.

Topics

Algorithms and Complexity Bioinformatics 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arnborg, S., Lagergren, J., Seese, D.: Easy problems for tree-decomposable graphs. J. Algorithms 12, 308–340 (1991)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Alon, N., Yuster, R., Zwick, U.: Color-coding. Journal of the ACM 42, 844–856 (1995)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bodlaender, H.L., Fellows, M., Langston, M., Ragan, M.A., Rosamond, F., Weyer, M.: Quadratic kernelization for convex recoloring of trees. In: Proceedings COCOON 2007, these proceedings (2007)Google Scholar
  4. 4.
    Bar-Yehuda, R., Feldman, I., Rawitz, D.: Improved approximation algorithm for convex recoloring of trees. In: Erlebach, T., Persinao, G. (eds.) WAOA 2005. LNCS, vol. 3879, pp. 55–68. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. 5.
    Borie, R.B., Parker, R.G., Tovey, C.A.: Automatic generation of linear-time algorithms from predicate calculus descriptions of problems on recursively generated graph families. Algorithmica 7, 555–581 (1992)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Bodlaender, H.L., Weyer, M.: Convex anc connected recolourings of trees and graphs. Manuscript (2005)Google Scholar
  7. 7.
    Bu, D., Zhao, Y., Cai, L., Xue, H., Zhu, X., Lu, H., Zhang, J., Sun, S., Ling, L., Zhang, N., Li, G., Chen, R.: Topological structure analysis of the protein-protein interaction network in budding yeast. Nucleic Acids Res. 31(9), 2443–2450 (2003)CrossRefGoogle Scholar
  8. 8.
    Chen, J., Chor, B., Fellows, M., Huang, X., Juedes, D., Kanj, I., Xia, G.: Tight lower bounds for certain parameterized NP-hard problems. In: Proceedings of the 19th Annual IEEE Conference on Computational Complexity, pp. 150–160. IEEE Computer Society Press, Los Alamitos (2004)CrossRefGoogle Scholar
  9. 9.
    Courcelle, B.: The monadic second-order logic of graphs I: Recognizable sets of finite graphs. Information and Computation 85, 12–75 (1990)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)Google Scholar
  11. 11.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)Google Scholar
  12. 12.
    Fellows, M., Giannopoulos, P., Knauer, C., Paul, C., Rosamond, F., Whitesides, S., Yu, N.: The lawnmower and other problems: applications of MSO logic in geometry, Manuscript (2007)Google Scholar
  13. 13.
    Gramm, J., Nickelsen, A., Tantau, T.: Fixed-parameter algorithms in phylogenetics. Manuscript (2006)Google Scholar
  14. 14.
    Huson, D.H., Bryant, D.: Application of phylogenetic networks in evolutionary studies. Mol. Biol. E 23, 254–267 (2006)CrossRefGoogle Scholar
  15. 15.
    Huson, D.H.: SplitsTree: a program for analyzing and visualizing evolutionary data. Bioinfomatics 14, 68–73 (1998)CrossRefGoogle Scholar
  16. 16.
    Kelley, B.P., Sharan, R., Karp, R.M., Sittler, T., Root, D.E., Stockwell, B.R., Ideker, T.: Conserved pathways within bacteria and yeast as revealed by global protein network alignment. Proc. Natl. Acad. Sci. USA 100, 11394–11399 (2003)CrossRefGoogle Scholar
  17. 17.
    Moran, S., Snir, S.: Convex recolorings of strings and trees: definitions, hardness results and algorithms. To appear in Journal of Computer and System Sciences. In: Dehne, F., López-Ortiz, A., Sack, J.-R. (eds.) WADS 2005. LNCS, vol. 3608, pp. 218–232. Springer, Heidelberg (2005) A preliminary version appearedGoogle Scholar
  18. 18.
    Moran, S., Snir, S., Sung, W.: Partial convex recolorings of trees and galled networks. Manuscript (2006)Google Scholar
  19. 19.
    Niedermeier, R.: Invitation to Fixed Parameter Algorithms. Oxford University Press, Oxford (2006)MATHGoogle Scholar
  20. 20.
    Ramadan, E., Tarafdar, A., Pothen, A.: A hypergraph model for the yeast protein complex network. In: Fourth IEEE International Workshop on High Performance Computational Biology, Santa Fe, NM, April 26, 2004. IEEE Computer Society Press, Los Alamitos (2004)Google Scholar
  21. 21.
    Rual, J.F., Venkatesan, K., Hao, T., Hirozane-Kishikawa, T., Dricot, A., Li, N., Berriz, G.F., Gibbons, F.D., Dreze, M., Ayivi-Guedehoussou, N., Klitgord, N., Simon, C., Boxem, M., Milstein, S., Rosenberg, J., Goldberg, D.S., Zhang, L.V., Wong, S.L., Franklin, G., Li, S., Albala, J.S., Lim, J., Fraughton, C., Llamosas, E., Cevik, S., Bex, C., Lamesch, P., Sikorski, R.S., Vandenhaute, J., Zoghbi, H.Y., Smolyar, A., Bosak, S., Sequerra, R., Doucette-Stamm, L., Cusick, M.E., Hill, D.E., Roth, F.P., Vidal, M.: Nature 437, 1173–1178 (2005)Google Scholar
  22. 22.
    Schwikowski, B., Uetz, P., Fields, S.: A network of protein-protein interactions in yeast. Nature Biotechnology 18(12), 1257–1261 (2000)CrossRefGoogle Scholar
  23. 23.
    Viveshwara, S., Brinda, K.V., Kannan, N.: Protein structure: insights from graph theory. J. Theoretical and Computational Chemistry 1, 187–211 (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Benny Chor
    • 1
  • Michael Fellows
    • 2
    • 3
  • Mark A. Ragan
    • 4
  • Igor Razgon
    • 5
  • Frances Rosamond
    • 2
  • Sagi Snir
    • 6
  1. 1.Computer Science Department, Tel Aviv University, Tel AvivIsrael
  2. 2.University of Newcastle, Callaghan NSW 2308Australia
  3. 3.Durham University, Institute of Advanced Study, Durham DH1 3RLUnited Kingdom
  4. 4.Institute for Molecular Biosciences, University of Queensland, Brisbane, QLD 4072Australia
  5. 5.Computer Science Department, University College CorkIreland
  6. 6.Department of Mathematics, University of California, BerkeleyUSA

Personalised recommendations