Abstract
In many network applications, one searches for a connected subset of vertices that exhibits other desirable properties. To this end, this paper studies the connected subgraph polytope of a graph, which is the convex hull of subsets of vertices that induce a connected subgraph. Much of our work is devoted to the study of two nontrivial classes of valid inequalities. The first are the a, b-separator inequalities, which have been successfully used to enforce connectivity in previous computational studies. The second are the indegree inequalities, which have previously been shown to induce all nontrivial facets for trees. We determine the precise conditions under which these inequalities induce facets and when each class fully describes the connected subgraph polytope. Both classes of inequalities can be separated in polynomial time and admit compact extended formulations. However, while the a, b-separator inequalities can be lifted in linear time, it is NP-hard to lift the indegree inequalities.
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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, Berlin (2013)
Álvarez-Miranda, E., Ljubić, I., Mutzel, P.: The rooted maximum node-weight connected subgraph problem. In: Gomes, C., Sellmann, M. (eds.) Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems, Lecture Notes in Computer Science, vol. 7874, pp. 300–315. Springer, Berlin (2013)
Backes, C., Rurainski, A., Klau, G., Müller, O., Stöckel, D., Gerasch, A., Küntzer, J., Maisel, D., Ludwig, N., Hein, M., et al.: An integer linear programming approach for finding deregulated subgraphs in regulatory networks. Nucl. Acids Res. 40(6), e43 (2012)
Bailly-Bechet, M., Borgs, C., Braunstein, A., Chayes, J., Dagkessamanskaia, A., François, J.M., Zecchina, R.: Finding undetected protein associations in cell signaling by belief propagation. Proc. Nat. Acad. Sci. 108(2), 882–887 (2011)
Biha, M., Kerivin, H., Ng, P.: Polyhedral study of the connected subgraph problem. Discrete Math. 338(1), 80–92 (2015)
Buchanan, A., Sung, J., Butenko, S., Pasiliao, E.: An integer programming approach for fault-tolerant connected dominating sets. INFORMS J. Comput. 27(1), 178–188 (2015)
Carvajal, R., Constantino, M., Goycoolea, M., Vielma, J., Weintraub, A.: Imposing connectivity constraints in forest planning models. Oper. Res. 61(4), 824–836 (2013)
Chen, C.Y., Grauman, K.: Efficient activity detection with max-subgraph search. In: Computer Vision and Pattern Recognition (CVPR), 2012 IEEE Conference on, pp. 1274–1281. IEEE (2012)
Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173, 4th edn. Springer, Heidelberg (2010)
Dittrich, M., Klau, G., Rosenwald, A., Dandekar, T., Müller, T.: Identifying functional modules in protein-protein interaction networks: an integrated exact approach. Bioinformatics 24(13), i223–i231 (2008)
Du, D.Z., Wan, P.: Connected Dominating Set: Theory and Applications. Springer Optimization and Its Applications, vol. 77. Springer, New York (2013)
Eppstein, D.: When does a graph admit an orientation in which there is at most one \(s\)–\(t\) walk? Theoretical computer science stack exchange. http://cstheory.stackexchange.com/questions/31412/when-does-a-graph-admit-an-orientation-in-which-there-is-at-most-one-s-t-walk (version: 2015-05-07)
Fischetti, M., Leitner, M., Ljubić, I., Luipersbeck, M., Monaci, M., Resch, M., Salvagnin, D., Sinnl, M.: Thinning out Steiner trees: a node-based model for uniform edge costs. Math. Prog. Comp. 1–27 (2016). doi:10.1007/s12532-016-0111-0
Garfinkel, R., Nemhauser, G.: Optimal political districting by implicit enumeration techniques. Manag. Sci. 16(8), B-495 (1970)
Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization, Algorithms and Combinatorics, vol. 2, 2nd edn. Springer, Berlin (1993)
Johnson, D.: The NP-completeness column: an ongoing guide. J. Algorithms 6(1), 145–159 (1985)
Korte, B., Lovász, L., Schrader, R.: Greedoids, Algorithms and Combinatorics, vol. 4. Springer, Berlin (1991)
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. 105(2–3), 427–449 (2006)
Lucena, A., Resende, M.: Strong lower bounds for the prize collecting Steiner problem in graphs. Discrete Appl. Math. 141(1), 277–294 (2004)
Moody, J., White, D.: Structural cohesion and embeddedness: a hierarchical concept of social groups. Am. Sociol. Rev. 68(1), 103–127 (2003)
Nemhauser, G., Wolsey, L.: Integer and Combinatorial Optimization, Discrete Mathematics and Optimization, vol. 18. Wiley, New York (1988)
Papadimitriou, C., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Dover Publications, Mineola (1998)
Raghavan, S., Magnanti, T.: Network connectivity. In: Dell’Amico, M., Maffioli, F., Martello, S. (eds.) Annotated Bibliographies in Combinatorial Optimization, pp. 335–354. Wiley, New York (1997)
Rajagopalan, S., Vazirani, V.: On the bidirected cut relaxation for the metric Steiner tree problem. SODA 99, 742–751 (1999)
Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency, Algorithms and Combinatorics, vol. 24. Springer, Berlin (2003)
Vijayanarasimhan, S., Grauman, K.: Efficient region search for object detection. In: Computer Vision and Pattern Recognition (CVPR), 2011 IEEE Conference on, pp. 1401–1408. IEEE (2011)
Wasserman, S., Faust, K.: Social Network Analysis: Methods and Applications, Strutural Analysis in the Social Sciences, vol. 8. Cambridge University Press, New York (1994)
Acknowledgements
We thank very much the anonymous referee who introduced us to indegree inequalities and the book of Korte et al. [17]. This material is based upon work supported by the AFRL Mathematical Modeling and Optimization Institute. Partial support by AFOSR under Grants FA9550-12-1-0103 and FA8651-12-2-0011 and by NSF under Grant CMMI-1538493 is gratefully acknowledged.
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Wang, Y., Buchanan, A. & Butenko, S. On imposing connectivity constraints in integer programs. Math. Program. 166, 241–271 (2017). https://doi.org/10.1007/s10107-017-1117-8
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DOI: https://doi.org/10.1007/s10107-017-1117-8
Keywords
- Connected subgraph polytope
- Maximum-weight connected subgraph
- Connectivity constraints
- Prize-collecting Steiner tree
- Contiguity