Abstract
Finding differentially regulated subgraphs in a biochemical network is an important problem in bioinformatics. We present a new model for finding such subgraphs which takes the polarity of the edges (activating or inhibiting) into account, leading to the problem of finding a connected subgraph induced by \(k\) vertices with maximum weight. We present several algorithms for this problem, including dynamic programming on tree decompositions and integer linear programming. We compare the strength of our integer linear program to previous formulations of the \(k\)-cardinality tree problem. Finally, we compare the performance of the algorithms and the quality of the results to a previous approach for finding differentially regulated subgraphs.
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References
Arora, S., Karakostas, G.: A 2 + \(\epsilon \) approximation algorithm for the k-MST problem. In: Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, SODA ’00, pp. 754–759. Society for Industrial and Applied Mathematics, Philadelphia (2000)
Álvarez Miranda, E., Ljubić, I., Mutzel, P.: The maximum weight connected subgraph problem. In: Jünger, M., Reinelt, G. (eds.) Facets of Combinatorial Optimization, pp. 245–270. Springer, Heidelberg (2013)
Blum, C., Blesa, M.J.: New metaheuristic approaches for the edge-weighted K-cardinality tree problem. Comput. Oper. Res. 32(6), 1355–1377 (2005)
Bhaskara, A., Charikar, M., Chlamtac, E., Feige, U., Vijayaraghavan, A.: Detecting high log-densities - an \({O}(n^{1/4})\) approximation for densest \(k\)-subgraph. In: Proceedings of the Forty-second ACM Symposium on Theory of Computing, STOC ’10, pp. 201–210. ACM, New York (2010)
Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25(6), 1305–1317 (1996)
Backes, C., Rurainski, A., Klau, G.W., Müller, O., Stöckel, D., Gerasch, A., Küntzer, J., Maisel, D., Ludwig, N., Hein, M., Keller, A., Burtscher, H., Kaufmann, M., Meese, E., Lenhof, H.-P.: An integer linear programming approach for finding deregulated subgraphs in regulatory networks. Nucleic Acids Res. (2011)
Blum, A., Ravi, R., Vempala, S.: A constant-factor approximation algorithm for the k-MST problem. J. Comput. Syst. Sci. 58(1), 101–108 (1999)
Chimani, M., Kandyba, M., Ljubić, I., Mutzel, P.: Obtaining optimal K-cardinality trees fast. J. Exp. Algorithmics 14, 5:2.5–5:2.23 (2010)
Croft, D., Mundo, A.F., Haw, R., Milacic, M., Weiser, J., Wu, G., Caudy, M., Garapati, P., Gillespie, M., Kamdar, M.R., Jassal, B., Jupe, S., Matthews, L., May, B., Palatnik, S., Rothfels, K., Shamovsky, V., Song, H., Williams, M., Birney, E., Hermjakob, H., Stein, L., D’Eustachio, P.: The reactome pathway knowledgebase. Nucleic Acids Res. 42(D1), D472–D477 (2014)
Chimani, M., Mutzel, P., Zey, B.: Improved Steiner tree algorithms for bounded treewidth. J. Discrete Algorithms 16, 67–78 (2012)
Cohen, N.: Several graph problems and their LP formulation (explanatory supplement for the Sage graph library), July 2010. http://hal.inria.fr/inria-00504914
Dittrich, M.T., Klau, G.W., Rosenwald, A., Dandekar, T., Müller, T.: Identifying functional modules in protein–protein interaction networks: an integrated exact approach. Bioinformatics 24(13), 223–231 (2008)
Dinu, I., Potter, J.D., Mueller, T., Liu, Q., Adewale, A.J., Jhangri, G.S., Einecke, G., Famulski, K.S., Halloran, P., Yasui, Y.: Gene-set analysis and reduction. Briefings Bioinf. 10, 24–34 (2009)
Fix, A., Chen, J., Boros, E., Zabih, R.: Approximate MRF inference using bounded treewidth subgraphs. In: Fitzgibbon, A., Lazebnik, S., Perona, P., Sato, Y., Schmid, C. (eds.) ECCV 2012, Part I. LNCS, vol. 7572, pp. 385–398. Springer, Heidelberg (2012)
Fischetti, M., Hamacher, H.W., Jørnsten, K., Maffioli, F.: Weighted k-cardinality trees: complexity and polyhedral structure. Networks 24(1), 11–21 (1994)
Feige, U., Kortsarz, G., Peleg, D.: The dense k-subgraph problem. Algorithmica 29(3), 410–421 (2001)
Garg, N.: A 3-approximation for the minimum tree spanning k vertices. In: Proceedings of the 37th Annual Symposium on Foundations of Computer Science, pp. 302–309, October 1996
Geistlinger, L., Csaba, G., Küffner, R., Mulder, N., Zimmer, R.: From sets to graphs: towards a realistic enrichment analysis of transcriptomic systems. Bioinformatics 27(13), i366–i373 (2011)
Gurobi Optimization, Inc., Gurobi Optimizer Reference Manual (2014). http://www.gurobi.com/documentation/5.0/reference-manual/
Ideker, T., Ozier, O., Schwikowski, B., Siegel, A.F.: Discovering regulatory and signalling circuits in molecular interaction networks. Bioinformatics 18(Suppl. 1), S233–S240 (2002)
Khot, S.: Ruling out PTAS for graph min-bisection, dense k-subgraph, and bipartite clique. SIAM J. Comput. 36, 1025–1071 (2006)
Kleinberg, J., Tardos, E.: Algorithm Design. Addison-Wesley Longman Publishing Co. Inc., Boston (2005)
Lawler, E.: Combinatorial Optimization: Networks and Matroids. Holt, Rinehart and Winston, New York (1976)
Ljubić, I.: Exact and memetic algorithms for two network design problems. Ph.D. thesis, Technische Universität Wien (2004). https://www.ads.tuwien.ac.at/publications/bib/pdf/ljubicPhD.pdf
Miller, C.E., Tucker, A.W., Zemlin, R.A.: Integer programming formulation of traveling salesman problems. J. ACM 7(4), 326–329 (1960)
Oracle corporation. Java Platform, Standard Edition 7 (2012). http://docs.oracle.com/javase/7/docs/api/
Quintão, F.P., da Cunha, A.S., Mateus, G.R., Lucena, A.: The k-cardinality tree problem: reformulations and lagrangian relaxation. Discrete Appl. Math. 158(12), 1305–1314 (2010)
Quintão, F.P., da Cunha, A.S., Mateus, G.R.: Integer programming formulations for the k-cardinality tree problem. Electron. Notes Discrete Math. 30, 225–230 (2008)
Subramanian, A., Tamayo, P., Mootha, V.K., Mukherjee, S., Ebert, B.L., Gillette, M.A., Paulovich, A., Pomeroy, S.L., Golub, T.R., Lander, E.S., Mesirov, J.P.: Gene set enrichment analysis: a knowledge-based approach for interpreting genome-wide expression profiles. Proc. Natl Acad. Sci. U.S.A. 102(43), 15545–15550 (2005)
Sathiamoorthy Subbarayan. TreeD: A Library for Tree Decomposition, July 2007. http://itu.dk/people/sathi/treed/
Zhao, X.-M.M., Wang, R.-S.S., Chen, L., Aihara, K.: Uncovering signal transduction networks from high-throughput data by integer linear programming. Nucleic Acids Res. 36(9), e48 (2008)
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Althaus, E., Blumenstock, M., Disterhoft, A., Hildebrandt, A., Krupp, M. (2014). Algorithms for the Maximum Weight Connected \(k\)-Induced Subgraph Problem. In: Zhang, Z., Wu, L., Xu, W., Du, DZ. (eds) Combinatorial Optimization and Applications. COCOA 2014. Lecture Notes in Computer Science(), vol 8881. Springer, Cham. https://doi.org/10.1007/978-3-319-12691-3_21
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