Abstract
This paper considers the Steiner tree problem in the model of two-stage stochastic optimization with recourse. This model, the focus of much recent research [11, 16, 8, 18], tries to capture the fact that many infrastructure planning problems have to be solved in the presence of uncertainty, and that we have make decisions knowing merely market forecasts (and not the precise set of demands); by the time the actual demands arrive, the costs may be higher due to inflation.
In the context of the Stochastic Steiner Tree problem on a graph G = (V,E), the model can be paraphrased thus: on Monday, we are given a probability distribution π on subsets of vertices, and can build some subset E M of edges. On Tuesday, a set of terminals D materializes (drawn from the same distribution π). We now have to buy edges E T so that the set E M ∪ E T forms a Steiner tree on D. The goal is to minimize the expected cost of the solution.
We give the first constant-factor approximation algorithm for this problem. To the best of our knowledge, this is the first O(1)-approximation for the stochastic version of a non sub-additive problem. In fact, algorithms for the unrooted stochastic Steiner tree problem we consider are powerful enough to solve the Multicommodity Rent-or-Buy problem, itself a topic of recent interest [3, 7, 15].
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Agrawal, A., Klein, P., Ravi, R.: When trees collide: an approximation algorithm for the generalized steiner problem on networks. SIAM J. Comput. 24(3), 440–456 (1995); Preliminary version in 23rd STOC (1991)
Beale, E.M.L.: On minimizing a convex function subject to linear inequalities. J. Roy. Statist. Soc. Ser. B. 17, 173–184; discussion, 194–203 (1955) (Symposium on linear programming)
Becchetti, L., Könemann, J., Leonardi, S., Pál, M.: Sharing the cost more efficiently: Improved approximation for multicommodity rent-or-buy. In: Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (2005)
Birge, J.R., Louveaux, F.: Introduction to stochastic programming. Springer Series in Operations Research. Springer, New York (1997)
Dantzig, G.B.: Linear programming under uncertainty. Management Sci. 1, 197–206 (1955)
Goemans, M.X., Williamson, D.P.: A general approximation technique for constrained forest problems. SIAM J. Comput. 24(2), 296–317 (1995); (Preliminary version in 5th SODA (1994))
Gupta, A., Kumar, A., Pál, M., Roughgarden, T.: Approximation via cost sharing: A simple approximation algorithm for the multicommodity rent or buy problem. In: Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, pp. 606–615 (2003)
Gupta, A., Pál, M., Ravi, R., Sinha, A.: Boosted sampling: Approximation algorithms for stochastic optimization. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing (2004)
Gupta, A., Ravi, R., Sinha, A.: An edge in time saves nine: Lp rounding approximation algorithms. In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science (2004)
Hayrapetyan, A., Swamy, C., Tardos, É.: Network design for information networks. In: ACM-SIAM Symposium on Discrete Algorithms (2005)
Immorlica, N., Karger, D., Minkoff, M., Mirrokni, V.: On the costs and benefits of procrastination: Approximation algorithms for stochastic combinatorial optimization problems. In: Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms (2004)
Kall, P., Wallace, S.W.: Stochastic programming. Wiley-Interscience Series in Systems and Optimization. John Wiley & Sons Ltd., Chichester (1994)
Karger, D.R., Minkoff, M.: Building steiner trees with incomplete global knowledge. In: Proceedings of the 41st Annual Symposium on Foundations of Computer Science, pp. 613–623 (2000)
Könemann, J., Leonardi, S., Schäffer, G.: A group-strategyproof mechanism for steiner forests. In: Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (2005)
Kumar, A., Gupta, A., Roughgarden, T.: A constant factor approximation algorithm for the multicommodity rent-or-buy problem. In: Proceedings of the 43rd Annual Symposium on Foundations of Computer Science (2002)
Ravi, R., Sinha, A.: Hedging uncertainty: Approximation algorithms for stochastic optimization problems. In: Proceedings of the 10th International Conference on Integer Programming and Combinatorial Optimization, IPCO (2004); GSIA Working Paper 2003-E68
Schultz, R., Stougie, L., van der Vlerk, M.H.: Two-stage stochastic integer programming: a survey. Statist. Nederlandica 50(3), 404–416 (1996)
Shmoys, D., Swamy, C.: Stochastic optimization is (almost) as easy as deterministic optimization. In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science (2004)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gupta, A., Pál, M. (2005). Stochastic Steiner Trees Without a Root. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds) Automata, Languages and Programming. ICALP 2005. Lecture Notes in Computer Science, vol 3580. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11523468_85
Download citation
DOI: https://doi.org/10.1007/11523468_85
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-27580-0
Online ISBN: 978-3-540-31691-6
eBook Packages: Computer ScienceComputer Science (R0)