On the Minimal Steiner Tree Subproblem and Its Application in Branch-and-Price
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.
Keywordsnetworks preflow-push algorithms branch-and-cut Steiner trees
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- 1.Ahuja, R.K., Magnanti, T.L., Orlin, J.B.: Network flows: theory, algorithms and applications. Prentice-Hall, New Jersey (1993)Google Scholar
- 6.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)Google Scholar
- 7.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)Google Scholar
- 13.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)Google Scholar
- 14.Levine, M.S.: Experimental study of minimum cut algorithms. Master’s thesis, Massachusetts Institute of Technology (1995)Google Scholar
- 17.Panagiotidi, S.: Efficient implementation of a preflow push solver and its application to network design. Master’s thesis, Imperial College London (September 2004)Google Scholar
- 18.Polzin, T.: Algorithms for the Steiner Problem in Networks. PhD thesis, Faculty of Science and Technology, University of Saarlandes (May 2003)Google Scholar