Abstract
The minimal Steiner tree problem is a classical NP-complete problem that has several applications in the communication and transportation sectors. It has recently emerged as a subproblem in decomposition techniques such as column generation and Lagrangian schemes. This has set new computational challenges to the state of the art solving approaches. Our goal is to improve on existing branch-and-cut algorithms so that our approach successfully serves as a fast subproblem solver in a decomposition context. Compared with existing literature, our technical contributions include 1) a new preflow-push cutting strategy, revisiting a little known graph algorithm, that halves the runtime of the separation step, and 2) a branching scheme that fairly balances the search tree and speeds up the search. An evaluation in a multicast design application shows that the algorithm enhances a column generation hybrid. Moreover, our approach offers a significant speedup factor on a publicly available set of challenging Steiner tree benchmarks.
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References
Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network flows: theory, algorithms and applications. Prentice-Hall, New Jersey (1993)
Cherkassky, B.V., Goldberg, A.V.: On implementing the push-relabel method for the maximum flow problem. Algorithmica 19, 390–410 (1997)
Chopra, S., Gorres, E.R., Rao, M.R.: Solving the Steiner tree problem on a graph using branch and cut. Journal on Computing 4(3), 320–335 (1992)
Chopra, S., Rao, M.R.: The Steiner tree problem I: Formulations, compositions and extension of facets. Mathematical Programming 64, 209–229 (1994)
Cronholm, W., Ajili, F.: Strong cost-based filtering for Lagrange decomposition applied to network design. In: Wallace, M. (ed.) CP 2004. LNCS, vol. 3258, pp. 726–730. Springer, Heidelberg (2004)
Cronholm, W., Ajili, F.: Hybrid branch-and-price for multicast network design. In: Proceedings of the 2nd International Network Optimization Conference (March 2005) (to appear)
Cronholm, W., Ouaja, W., Ajili, F.: Strengthening optimality reasoning for a network routing application. In: Proceedings of the Fourth Workshop on Cooperative Solvers in Constraint Programming, COSOLV 2004, Toronto, Canada (September 2004)
Goldberg, A.V., Tarjan, R.E.: A new approach to the maximum-flow problem. Journal of the Association for Computing Machinery 35(4), 921–940 (1988)
Hao, J., Orlin, J.B.: A faster algorithm for finding the minimum cut in a directed graph. Journal of Algorithms 17, 424–446 (1994)
Hwang, F.K., Richards, D.S., Winter, P.: The Steiner Tree Problem. Annals of Discrete Mathematics, vol. 53. Elsevier Science Publishers B. V., Amsterdam (1992)
Kerivin, H., Mahjoub, A.R.: Separation of partition inequalities for the (1,2)-survivable network design problem. Operations Research Letters 30(4), 265–268 (2002)
Koch, T., Martin, A.: Solving Steiner tree problems in graphs to optimality. Networks 32(3), 207–232 (1998)
Koch, T., Martin, A., Voß, S.: SteinLib: An updated library on steiner tree problems in graphs. Technical Report ZIB-Report 00-37, Konrad-Zuse-Zentrum für Informationstechnik Berlin, Takustr. 7, Berlin (2000)
Levine, M.S.: Experimental study of minimum cut algorithms. Master’s thesis, Massachusetts Institute of Technology (1995)
Lucena, A., Beasley, J.E.: A branch and cut algorithm for the Steiner problem in graphs. Networks, an International Journal 31(1), 39–59 (1998)
Padberg, M., Rinaldi, G.: An efficient algorithm for the minimum capacity cut problem. Mathematical Programming 47, 19–36 (1990)
Panagiotidi, S.: Efficient implementation of a preflow push solver and its application to network design. Master’s thesis, Imperial College London (September 2004)
Polzin, T.: Algorithms for the Steiner Problem in Networks. PhD thesis, Faculty of Science and Technology, University of Saarlandes (May 2003)
Prytz, M., Forsgren, A.: Dimensioning multicast-enabled communications networks. Networks 39(4), 216–231 (2002)
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Cronholm, W., Ajili, F., Panagiotidi, S. (2005). On the Minimal Steiner Tree Subproblem and Its Application in Branch-and-Price. In: Barták, R., Milano, M. (eds) Integration of AI and OR Techniques in Constraint Programming for Combinatorial Optimization Problems. CPAIOR 2005. Lecture Notes in Computer Science, vol 3524. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11493853_11
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DOI: https://doi.org/10.1007/11493853_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-26152-0
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