Abstract
We consider the Steiner tree problem under a 2-stage stochastic model with recourse and finitely many scenarios (SSTP). Thereby, edges are purchased in the first stage when only probabilistic information on the set of terminals and the future edge costs is known. In the second stage, one of the given scenarios is realized and additional edges are purchased to interconnect the set of (now known) terminals. The goal is to choose an edge set to be purchased in the first stage while minimizing the overall expected cost of the solution.
We provide a new semi-directed cut-set based integer programming formulation that is stronger than the previously known undirected model. To solve the formulation to provable optimality, we suggest a two-stage branch-and-cut framework, facilitating (integer) L-shaped cuts. The framework itself is also applicable to a range of other stochastic problems.
As SSTP has yet been investigated only from the theoretical point of view, we also present the first computational study for SSTP, showcasing the applicability of our approach and its benefits over solving the extensive form of the deterministic equivalent directly.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Benders, J.F.: Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik 4, 238–252 (1962)
Birge, J.R., Louveaux, F.: A multicut algorithm for two-stage stochastic linear programs. European Journal of Operational Research 34, 384–392 (1988)
Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer, New York (1997)
Byrka, J., Grandoni, F., Rothvoß, T., Sanità, L.: An improved LP-based approximation for Steiner tree. In: ACM STOC (to appear, 2010)
Carøe, C.C., Tind, J.: L-shaped decomposition of two-stage stochastic programs with integer recourse. Mathematical Programming 83(3), 451–464 (1998)
Goemans, M.X., Williamson, D.P.: The primal-dual method for approximation algorithms and its application to network design problems. In: Approximation algorithms for NP-hard problems, pp. 144–191 (1996)
Gupta, A., Hajiaghayi, M., Kumar, A.: Stochastic Steiner tree with non-uniform inflation. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds.) RANDOM 2007 and APPROX 2007. LNCS, vol. 4627, pp. 134–148. Springer, Heidelberg (2007)
Gupta, A., Ravi, R., Sinha, A.: LP rounding approximation algorithms for stochastic network design. Math. of Operations Research 32(2), 345–364 (2007)
Johnson, D.S., Minkoff, M., Phillips, S.: The prize-collecting Steiner tree problem: Theory and practice. In: ACM-SIAM SODA, pp. 760–769. SIAM, Philadelphia (2000)
Laporte, G., Louveaux, F.: The integer L-shaped method for stochastic integer programs with complete recourse. Oper. Res. Lett. 13, 133–142 (1993)
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. Mathematical Programming 105(2-3), 427–449 (2006)
PCSTP Benchmark, http://homepage.univie.ac.at/ivana.ljubic/research/pcstp/
Shapiro, A., Ruszczynski, A.P., Dentcheva, D.: Lectures on Stochastic Programming: Modeling and Theory. SIAM, Philadelphia (2009)
SteinLib, http://steinlib.zib.de/steinlib.php
Swamy, C., Shmoys, D.: Approximation algorithms for 2-stage stochastic optimization problems 1(37), 33–46 (2006)
Verweij, B., Ahmed, S., Kleywegt, A.J., Nemhauser, G., Shapiro, A.: The sample average approximation method applied to stochastic routing problems: A computational study. Computational Optimization and Appl. 24(2-3), 289–333 (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bomze, I., Chimani, M., Jünger, M., Ljubić, I., Mutzel, P., Zey, B. (2010). Solving Two-Stage Stochastic Steiner Tree Problems by Two-Stage Branch-and-Cut. In: Cheong, O., Chwa, KY., Park, K. (eds) Algorithms and Computation. ISAAC 2010. Lecture Notes in Computer Science, vol 6506. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-17517-6_38
Download citation
DOI: https://doi.org/10.1007/978-3-642-17517-6_38
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-17516-9
Online ISBN: 978-3-642-17517-6
eBook Packages: Computer ScienceComputer Science (R0)