Abstract
We study the NP-complete Graph Motif problem: given a vertex-colored graph G = (V,E) and a multiset M of colors, does there exist an S ⊆ V such that G[S] is connected and carries exactly (also with respect to multiplicity) the colors in M? We present an improved randomized algorithm for Graph Motif with running time O(4.32|M|·|M|2·|E|). We extend our algorithm to list-colored graph vertices and the case where the motif G[S] needs not be connected. By way of contrast, we show that extending the request for motif connectedness to the somewhat “more robust” motif demands of biconnectedness or bridge-connectedness leads to W[1]-complete problems. Actually, we show that the presumably simpler problems of finding (uncolored) biconnected or bridge-connected subgraphs are W[1]-complete with respect to the subgraph size. Answering an open question from the literature, we further show that the parameter “number of connected motif components” leads to W[1]-hardness even when restricted to graphs that are paths.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alm, E., Arkin, A.P.: Biological networks. Curr. Opin. Struc. Biol. 13(2), 193–202 (2003)
Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)
Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Fourier meets Möbius: fast subset convolution. In: Proc. 39th STOC, pp. 67–74. ACM, New York (2007)
Cesati, M.: Perfect code is W[1]-complete. Inform. Process. Lett. 81, 163–168 (2002)
Deshpande, P., Barzilay, R., Karger, D.R.: Randomized decoding for selection-and-ordering problems. In: Proc. NAACL HLT 2007. Association for Computational Linguistics, pp. 444–451 (2007)
Dondi, R., Fertin, G., Vialette, S.: Weak pattern matching in colored graphs: Minimizing the number of connected components. In: Proc. 10th ICTCS. WSPC, vol. 4596, pp. 27–38. World Scientific, Singapore (2007)
Dost, B., Shlomi, T., Gupta, N., Ruppin, E., Bafna, V., Sharan, R.: QNet: A tool for querying protein interaction networks. In: Speed, T., Huang, H. (eds.) RECOMB 2007. LNCS (LNBI), vol. 4453, pp. 1–15. Springer, Heidelberg (2007)
Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, Heidelberg (1999)
Fellows, M.R., 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)
Hüffner, F., Wernicke, S., Zichner, T.: Algorithm engineering for color-coding to facilitate signaling pathway detection. In: Proc. 5th APBC. Advances in Bioinf. and Comput. Biol., vol. 5, pp. 277–286. Imperial College Press (2007); Extended version to appear in Algorithmica
Hüffner, F., Wernicke, S., Zichner, T.: FASPAD: fast signaling pathway detection. Bioinformatics 23(13), 1708–1709 (2007)
Lacroix, V., Fernandes, C.G., Sagot, M.-F.: Reaction motifs in metabolic networks. In: Casadio, R., Myers, G. (eds.) WABI 2005. LNCS (LNBI), vol. 3692, pp. 178–191. Springer, Heidelberg (2005)
Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, Oxford (2006)
Scott, J., Ideker, T., Karp, R.M., Sharan, R.: Efficient algorithms for detecting signaling pathways in protein interaction networks. J. Comput. Biol. 13(2), 133–144 (2006)
Sharan, R., Ideker, T.: Modeling cellular machinery through biological network comparison. Nat. Biotechnol. 24, 427–433 (2006)
Shen-Orr, S., Milo, R., Mangan, S., Alon, U.: Network motifs in the transcriptional regulation network of escherichia coli. Nat. Genet. 31(1), 64–68 (2002)
Tarjan, R.E.: Depth first search and linear graph algorithms. SIAM J. Comp. (1), 146–160 (1972)
Wernicke, S.: Efficient detection of network motifs. IEEE ACM T. Comput. Bi. 3(4), 347–359 (2006)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Betzler, N., Fellows, M.R., Komusiewicz, C., Niedermeier, R. (2008). Parameterized Algorithms and Hardness Results for Some Graph Motif Problems. In: Ferragina, P., Landau, G.M. (eds) Combinatorial Pattern Matching. CPM 2008. Lecture Notes in Computer Science, vol 5029. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69068-9_6
Download citation
DOI: https://doi.org/10.1007/978-3-540-69068-9_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-69066-5
Online ISBN: 978-3-540-69068-9
eBook Packages: Computer ScienceComputer Science (R0)