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Finding Approximate and Constrained Motifs in Graphs

  • Riccardo Dondi
  • Guillaume Fertin
  • Stéphane Vialette
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6661)

Abstract

One of the emerging topics in the analysis of biological networks is the inference of motifs inside a network. In the context of metabolic network analysis, a recent approach introduced in [14], represents the network as a vertex-colored graph, while a motif \(\mathcal{M}\) is represented as a multiset of colors. An occurrence of a motif \(\mathcal{M}\) in a vertex-colored graph G is a connected induced subgraph of G whose vertex set is colored exactly as \(\mathcal{M}\). We investigate three different variants of the initial problem. The first two variants, Min-Add and Min-Substitute, deal with approximate occurrences of a motif in the graph, while the third variant, Constrained Graph Motif (or CGM for short), constrains the motif to contain a given set of vertices. We investigate the classical and parameterized complexity of the three problems. We show that Min-Add and Min-Substitute are NP-hard, even when \(\mathcal{M}\) is a set, and the graph is a tree of degree bounded by 4 in which each color appears at most twice. Moreover, we show that Min-Substitute is in FPT when parameterized by the size of \(\mathcal{M}\). Finally, we consider the parameterized complexity of the CGM problem, and we give a fixed-parameter algorithm for graphs of bounded treewidth, while we show that the problem is W[2]-hard, even if the input graph has diameter 2.

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References

  1. 1.
    Ambalath, A.M., Balasundaram, R., Rao H., C., Koppula, V., Misra, N., Philip, G., Ramanujan, M.S.: On the Kernelization Complexity of Colorful Motifs. In: Raman, V., Saurabh, S. (eds.) IPEC 2010. LNCS, vol. 6478, pp. 14–25. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  2. 2.
    Alimonti, P., Kann, V.: Some APX-Completeness Results for Cubic Graphs. Theor. Comput. Sci. 237(1-2), 123–134 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alon, N., Yuster, R., Zwick, U.: Color Coding. Journal of the ACM 42(4), 844–856 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Betzler, N., Fellows, M.R., Komusiewicz, C., Niedermeier, R.: Parameterized Algorithms and Hardness Results for Some Graph Motif Problems. In: Ferragina, P., Landau, G.M. (eds.) CPM 2008. LNCS, vol. 5029, pp. 31–43. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  5. 5.
    Bruckner, S., Hüffner, F., Karp, R.M., Shamir, R., Sharan, R.: Topology-Free Querying of Protein Interaction Networks. In: Batzoglou, S. (ed.) RECOMB 2009. LNCS, vol. 5541, pp. 74–89. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  6. 6.
    Cesati, M.: Compendium of parameterized problems, http://bravo.ce.uniroma2.it/home/cesati/research/compendium.pdf
  7. 7.
    Dondi, R., Fertin, G., Vialette, S.: Weak Pattern Matching in Colored Graphs: Minimizing the Number of Connected Components. In: Italiano, G.F., Moggi, E., Laura, L. (eds.) ICTCS 2007, pp. 27–38. World Scientific, Singapore (2007)Google Scholar
  8. 8.
    Dondi, R., Fertin, G., Vialette, S.: Maximum Motif Problem in Vertex-Colored Graphs. In: Kucherov, G., Ukkonen, E. (eds.) CPM 2009 Lille. LNCS, vol. 5577, pp. 221–235. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  9. 9.
    Downey, R., Fellows, M.: Parameterized Complexity. Springer, Heidelberg (1999)CrossRefzbMATHGoogle Scholar
  10. 10.
    Fellows, M., Fertin, G., Hermelin, D., Vialette, S.: Sharp Tractability Borderlines for Finding Connected Motifs in Vertex-Colored Graphs. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 340–351. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  11. 11.
    Guillemot, S., Sikora, F.: Finding and Counting Vertex-Colored Subtrees. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 405–416. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  12. 12.
    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. Nat. Acad. Sci. 100(20), 11394–11399 (2003)CrossRefGoogle Scholar
  13. 13.
    Koyutürk, M., Grama, A., Szpankowski, W.: Pairwise Local Alignment of Protein Interaction Networks Guided by Models of Evolution. In: Miyano, S., Mesirov, J., Kasif, S., Istrail, S., Pevzner, P.A., Waterman, M. (eds.) RECOMB 2005. LNCS (LNBI), vol. 3500, pp. 48–65. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  14. 14.
    Lacroix, V., Fernandes, C.G., Sagot, M.F.: Motif Search in Graphs: Application to Metabolic Networks. IEEE/ACM Transactions on Computational Biology and Bioinformatics (TCBB) 3(4), 360–368 (2006)CrossRefGoogle Scholar
  15. 15.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)CrossRefzbMATHGoogle Scholar
  16. 16.
    Paz, A., Moran, S.: Non Deterministic Polynomial Optimization Problems and Their Approximations. Theor. Comput. Sci. 15, 251–277 (1981)CrossRefzbMATHGoogle Scholar
  17. 17.
    Scott, J., Ideker, T., Karp, R.M., Sharan, R.: Efficient Algorithms for Detecting Signaling Pathways in Protein Interaction Networks. Journal of Computational Biology 13, 133–144 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Sharan, R., Ideker, T., Kelley, B., Shamir, R., Karp, R.M.: Identification of Protein Complexes by Comparative Analysis of Yeast and Bacterial Protein Interaction Data. In: Bourne, P.E., Gusfield, D. (eds.) RECOMB 2004, pp. 282–289. ACM Press, New York (2004)Google Scholar
  19. 19.
    Sharan, R., Suthram, S., Kelley, R., Kuhn, T., McCuine, S., Uetz, P., Sittler, K.R.M., Ideker, T.: Conserved Patterns of Protein Interaction in Multiple Species. Proc. Nat. Acad. Sci. 102(6), 1974–1979 (2005)CrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Riccardo Dondi
    • 1
  • Guillaume Fertin
    • 2
  • Stéphane Vialette
    • 3
  1. 1.Dipartimento di Scienze dei Linguaggi, della Comunicazione e degli Studi CulturaliUniversità degli Studi di BergamoBergamoItaly
  2. 2.Laboratoire d’Informatique de Nantes-Atlantique (LINA), UMR CNRS 6241Université de NantesNantes Cedex 3France
  3. 3.IGM-LabInfo, CNRS UMR 8049Université Paris-EstMarne-la-ValléeFrance

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