The Maximum Weight Connected Subgraph Problem

  • Eduardo Álvarez-Miranda
  • Ivana Ljubić
  • Petra Mutzel

Abstract

The Maximum (Node-) Weight Connected Subgraph Problem (MWCS) searches for a connected subgraph with maximum total weight in a node-weighted (di)graph. In this work we introduce a new integer linear programming formulation built on node variables only, which uses new constraints based on node-separators. We theoretically compare its strength to previously used MIP models in the literature and study the connected subgraph polytope associated with our new formulation. In our computational study we compare branch-and-cut implementations of the new model with two models recently proposed in the literature: one of them using the transformation into the Prize-Collecting Steiner Tree problem, and the other one working on the space of node variables only. The obtained results indicate that the new formulation outperforms the previous ones in terms of the running time and in terms of the stability with respect to variations of node weights.

References

  1. 1.
    Backes, C., Rurainski, A., Klau, G., Müller, O., Stöckel, D., Gerasch, A., Küntzer, J., Maisel, D., Ludwig, N., Hein, M., Keller, A., Burtscher, H., Kaufmann, M., Meese, E., Lenhof, H.: An integer linear programming approach for finding deregulated subgraphs in regulatory networks. Nucleic Acids Res. 1, 1–13 (2011) Google Scholar
  2. 2.
    Bateni, M., Chekuri, C., Ene, A., Hajiaghayi, M., Korula, N., Marx, D.: Prize-collecting Steiner problems on planar graphs. In: Randall, D. (ed.) Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, CA, USA, January 23–25, pp. 1028–1049 (2011) Google Scholar
  3. 3.
    Carvajal, R., Constantino, M., Goycoolea, M., Vielma, J., Weintraub, A.: Imposing connectivity constraints in forest planning models. Oper. Res. (2013). doi:10.1287/opre.2013.1183 MATHGoogle Scholar
  4. 4.
    Chen, C.Y., Grauman, K.: Efficient activity detection with max-subgraph search. In: 2012 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Providence, RI, USA, June 16–21, pp. 1274–1281 (2012) CrossRefGoogle Scholar
  5. 5.
    Cherkassky, B.V., Goldberg, A.V.: On implementing push-relabel method for the maximum flow problem. Algorithmica 19, 390–410 (1994) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Chimani, M., Kandyba, M., Ljubic, I., Mutzel, P.: Obtaining optimal k-cardinality trees fast. ACM J. Exp. Algorithmics 14, 5 (2009) MathSciNetGoogle Scholar
  7. 7.
    Dilkina, B., Gomes, C.: Solving connected subgraph problems in wildlife conservation. In: Lodi, A., Milano, M., Toth, P. (eds.) CPAIOR. LNCS, vol. 6140, pp. 102–116. Springer, Berlin (2010) Google Scholar
  8. 8.
    Dittrich, M., Klau, G., Rosenwald, A., Dandekar, T., Müller, T.: Identifying functional modules in protein-protein interaction networks: an integrated exact approach. Bioinformatics 24, i223–i231 (2008) CrossRefGoogle Scholar
  9. 9.
    Feigenbaum, J., Papadimitriou, C.H., Shenker, S.: Sharing the cost of multicast transmissions. J. Comput. Syst. Sci. 63(1), 21–41 (2001) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Fischetti, M., Hamacher, H.W., Jørnsten, K., Maffioli, F.: Weighted k-cardinality trees: complexity and polyhedral structure. Networks 24(1), 11–21 (1994) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Fügenschuh, A., Fügenschuh, M.: Integer linear programming models for topology optimization in sheet metal design. Math. Methods Oper. Res. 68(2), 313–331 (2008) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    genetrail.bioinf.uni-sb.de/ilp/. Accessed 10 September 2012
  13. 13.
    Goldschmidt, O., Hochbaum, D.S.: k-edge subgraph problems. Discrete Appl. Math. 74(2), 159–169 (1997) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Grötschel, M.: Polyedrische Charakterisierungen Kombinatorischer Optimierungsprobleme. Mathematical Systems in Economics, vol. 36. Verlag Anton Hain, Meisenheim am Glan (1977) MATHGoogle Scholar
  15. 15.
    Grötschel, M., Monma, C.L.: Integer polyhedra arising from certain network design problems with connectivity constraints. SIAM J. Discrete Math. 3(4), 502–523 (1990) MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Grötschel, M., Monma, C.L., Stoer, M.: Facets for polyhedra arising in the design of communication networks with low-connectivity constraints. SIAM J. Optim. 2(3), 474–504 (1992) MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Grötschel, M., Monma, C.L., Stoer, M.: Polyhedral and computational investigations for designing communication networks with high survivability requirements. Oper. Res. 43(6), 1012–1024 (1995) MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Hochbaum, D.S., Pathria, A.: Node-optimal connected k-subgraphs. Manuscript, UC Berkeley (1994) Google Scholar
  19. 19.
    Ideker, T., Ozier, O., Schwikowski, B., Siegel, A.: Discovering regulatory and signalling circuits in molecular interaction networks. Bioinformatics 18(Suppl. 1), s233–s240 (2002) CrossRefGoogle Scholar
  20. 20.
    Johnson, D.S., Minkoff, M., Phillips, S.: The prize-collecting Steiner tree problem: theory and practice. In: Proceedings of the 11th ACM-SIAM Symposium on Discrete Algorithms, SODA 2000, San Francisco, CA, USA, 9–11 January, pp. 760–769 (2000) Google Scholar
  21. 21.
    Koch, T., Martin, A.: Solving Steiner tree problems in graphs to optimality. Networks 32, 207–232 (1998) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Lee, H., Dooly, D.R.: Algorithms for the constrained maximum-weight connected graph problem. Nav. Res. Logist. 43, 985–1008 (1996) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Lee, H., Dooly, D.: Decomposition algorithms for the maximum-weight connected graph problem. Nav. Res. Logist. 45, 817–837 (1998) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Ljubić, I., Weiskircher, R., Pferschy, U., Klau, G., Mutzel, P., Fischetti, M.: An algorithmic framework for the exact solution of the prize-collecting Steiner tree problem. Math. Program., Ser. B 105, 427–449 (2006) MATHCrossRefGoogle Scholar
  25. 25.
    www.planet-lisa.net/. Accessed 10 September 2012
  26. 26.
    Yamamoto, T., Bannai, H., Nagasaki, M., Miyano, S.: Better decomposition heuristics for the maximum-weight connected graph problem using betweenness centrality. In: Gama, J., Costa, V., Jorge, A., Brazdil, P. (eds.) Discovery Science. LNCS, vol. 5808, pp. 465–472. Springer, Berlin (2009) CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Eduardo Álvarez-Miranda
    • 1
  • Ivana Ljubić
    • 2
  • Petra Mutzel
    • 3
  1. 1.Dipartimento di Ingegneria dell’Energia Elettrica e dell’InformazioneUniversità di BolognaBolognaItaly
  2. 2.Institut für Statistik und Operations ResearchUniversität WienViennaAustria
  3. 3.Fakultät für InformatikTechnische Universität DortmundDortmundGermany

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