Abstract
Given an undirected graph \(G\) with vertex and edge weights, the \(k\)-cardinality tree problem asks for a minimum weight tree of \(G\) containing exactly \(k\) edges. In this paper we consider a directed graph reformulation of the problem and carry out a polyhedral investigation of the polytope defined by the convex hull of its feasible integral solutions. Three new families of valid inequalities are identified and two of them are proven to be facet implying for that polytope. Additionally, a Branch-and-cut algorithm that separates the new inequalities is implemented and computationally tested. The results obtained indicate that our algorithm outperforms its counterparts from the literature. Such a performance could be attributed, to a large extent, to the use of the new inequalities and also to some pre-processing tests introduced in this study.
Similar content being viewed by others
References
Awerbuch, B., Azar, Y., Blum, A., Vempala, S.: Improved approximation guarantees for minimum weight k-trees and prize-collecting salesmen. In: Proceedings of the 27th Annual ACM Symposium on Theory of Computing, ACM Press, pp. 277–283 (1995)
Beasley, J.E.: An SST-based Algorithm for the Steiner Problem in Graphs. Networks 19, 1–16 (1989)
Beasley, J.E.: OR-Library: distributing test problems by electronic mail. J. Oper. Res. Soc. 41(11), 1069–1072 (1990)
Blesa, M.J., Xhafa, F.: A C++ implementation of Tabu search for v-cardinality tree problem based on generic programming and component reuse. In. Net. Object. Days 2000, 648–652 (2000)
Blum, C.: Revisiting dynamic programming for finding optimal subtrees in trees. Eur. J. Oper. Res. 177, 102–115 (2007)
Blum, C., Blesa, M.J.: New metaheuristic approaches for the edge-weighted k-cardinality tree problem. Comput. Oper. Res. 32(6), 1355–1377 (2005)
Blum, C., Ehrgott, M.: Local search algorithms for the k-cardinality tree problem. Discret. Appl. Math. 128, 511–540 (2003)
Borndörfer, R., Ferreira, C., Martin, A.: Decomposing matrices into blocks. SIAM J. Optim. 9(1), 236–269 (1998)
Brimberg, J., Urosević, D., Mladenović, N.: Variable neighborhood search for the vertex weighted k-cardinality tree problem. Eur. J. Oper. Res. 171, 74–84 (2006)
Chimani, M., Kandyba, M., Ljubić, I., Mutzel, P.: Obtaining optimal \(k\) cardinality trees fast. ACM J. Exp. Algorithm., 14, 5:25–5:2.23 (2009)
da Cunha, A.S., Lucena, A., Maculan, N., Resende, M.: A relax-and-cut algorithm for the prize collecting steiner problem in graphs. Discret. Appl. Math. 157(6), 1198–1217 (2010)
Desrochers, M., Laporte, G.: Improvements and extensions to the miller-tucker-zemlin subtour elimination constraints. Oper. Res. Lett. 10, 27–36 (1991)
Duin, C.: Steiner’s Problem in Graphs. PhD thesis, University of Amsterdam (1993)
Ehrgott, M., Freitag, J.: \(\text{ K}\_\text{ TREE/K}\_\text{ SUBGRAPH}\): a program package for minimal weighted \(k\)-cardinality trees and subgraphs. Eur. J. Oper. Res. 93, 224–225 (1996)
Ehrgott, M., Hamacher, H.W., Freitag, J., Maffioli, F.: Heuristics for the k-cardinality tree and subgraph problems. Asia-Pacific J. Oper. Res. 14, 87–114 (1997)
Fischetti, M., Hamacher, H., Jörnsten, K., Maffioli, F.: Weighted k-cardinality trees: complexity and polyhedral structure. Networks 24, 11–21 (1994)
Foulds, L.R., Hamacher, H.W., Wilson, J.: Integer programming approaches to facilities layout models with forbidden areas. Ann. Oper. Res. 81, 405–417 (1998)
Garg, N., Hochbaum, D.: An o(log k) approximation algorithm for the k minimum spanning tree problem on the plane. Algorithmica 18, 111–121 (1997)
Jörnsten, K., Lokketangen, A.: Tabu search for weighted k-cardinality trees. Asia-Pacific J. Oper. Res. 14, 9–26 (1997)
Kandyba-Chimani, M.: Exact algorithms for Network Design Problems using Graph Orientations. PhD thesis, Fakultät für Informatik, Technischen Universität Dortmund, 2011. Available for download from: https://eldorado.tu-dortmund.de/bitstream/2003/27701/1/Dissertation.pdf
Kataoka, S., Araki, N., Yamada, T.: Upper and lower bounding procedures for minimum rooted \(k\)-subtree problem. Eur. J. Oper. Res. 122, 561–569 (2000)
Koch, T., Martin, A.: Solving steiner tree problems in graphs to optimality. Networks 32, 207–232 (1998)
Kruskal, J.B.: On the shortest spanning subtree of a graph and the traveling salesman problem. Proc. Am. Math. Soc. 7, 48–50 (1956)
Ljubić, I., Weiskircher, R., Pferschy, U., Klau, G., Mutzel, P., Fischetti, M.: Solving the Prize-Collecting Steiner Problem to Optimality. Technische Universität Wien, Institut für Computergraphik und Algorithmen, Technical Report (2004)
Lucena, A., Beasley, J.E.: A branch and cut algorithm for the steiner problem in graphs. Networks 31, 39–59 (1998)
Maculan, N.: The steiner problem in graphs. Ann. Discret. Math. 31, 185–212 (1987)
Marathe, M.V., Ravi, R., Ravi, S.S., Rosenkrantz, D.J., Sundaram, R.: Spanning trees—short or small. SIAM J. Discret. Math. 9(2), 178–200 (1996)
Miller, C.E., Tucker, A.W., Zemlin, R.A.: Integer programming formulations and travelling salesman problems. J. Assoc. Comput. Mach. 7, 326–329 (1960)
Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley Interscience, New York (1988)
Philpott, A.B., Wormald, N.C.: On the optimal extraction of minimum weight k trees. Technical Report TR-21/93., University of Auckland (1997)
Polzin, T., Daneshmand, S.V.: Improved algorithms for the steiner problem in networks. Discret. Appl. Math. 112(1–3), 263–300 (2001)
Prim, R.C.: Shortest connection networks and some generalizations. Bell Sys. Tech. J. 36, 1389–1401 (1957)
Quintão, F.P., da Cunha, A.S., Mateus, G.R., Lucena, A.: The \(k\)-cardinality tree problem: reformulations and lagrangian relaxation. Discret. Appl. Math. 158, 1305–1314 (2010)
Rosseti, I., Poggi de Aragão, M., Ribeiro, C.C., Uchoa, E., Werneck, R.F.: New benchmark instances for the Steiner problem in graphs. In: Extended Abstracts of the 4th Metaheuristics International Conference, pp. 557–561 (2001)
Simonetti L., Protti, F., Frota, Y., de Souza, C.C.: New branch-and-bound algorithms for k-cardinality tree problems. Elect. Notes Discret. Math. 37, 27–32 (2011). LAGOS’11 - VI Latin-American Algorithms, Graphs and Optimization Symposium
Acknowledgments
The authors wish to thank Ivana Ljubić and Maria Kandyba-Chimani for sending us detailed computational results of their algorithm.
Author information
Authors and Affiliations
Corresponding author
Additional information
Luidi Simonetti is partially funded by FAPERJ grant E-26/111.819/2010 and CNPq grants 483.243/2010-8, 304793/2011-6. Alexandre Salles da Cunha is partially funded by CNPq grants 302276/2009-2, 477863/2010-8 and FAPEMIG PRONEX APQ-01201-09. Abilio Lucena is partially funded by CNPq grant 310561/2009-4 and FAPERJ grant E26-110.552/2010.
Electronic supplementary material
Below is the link to the electronic supplementary material.
Rights and permissions
About this article
Cite this article
Simonetti, L., da Cunha, A.S. & Lucena, A. Polyhedral results and a Branch-and-cut algorithm for the \(k\)-cardinality tree problem. Math. Program. 142, 511–538 (2013). https://doi.org/10.1007/s10107-012-0590-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10107-012-0590-3