Abstract
We consider the problem of connecting given vertex pairs over a stochastic metric graph, each vertex of which has a probability of presence independently of all other vertices. Vertex pairs requiring connection are always present with probability 1. Our objective is to satisfy the connectivity requirements for every possibly materializable subgraph of the given metric graph, so as to optimize the expected total cost of edges used. This is a natural problem model for cost-efficient Steiner Forests on stochastic metric graphs, where uncertain availability of intermediate nodes requires fast adjustments of traffic forwarding. For this problem we allow a priori design decisions to be taken, that can be modified efficiently when an actual subgraph of the input graph materializes. We design a fast (almost linear time in the number of vertices) modification algorithm whose outcome we analyze probabilistically, and show that depending on the a priori decisions this algorithm yields 2 or 4 approximation factors of the optimum expected cost. We also show that our analysis of the algorithm is tight.
Research partially supported by the Greek Ministry of Education and Research under the project PYTHAGORAS II.
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
Agrawal, A., Klein, P.N., Ravi, R.: When Trees Collide: An Approximation Algorithm for the Generalized Steiner Problem on Networks. SIAM Journal on Computing 24(3), 440–456 (1995)
Bertsimas, D.: Probabilistic Combinatorial Optimization. PhD thesis (1988)
Bertsimas, D.: The probabilistic minimum spanning tree problem. Networks 20, 245–275 (1990)
Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. LNCS. Springer, Heidelberg (1997)
Dantzig, G.W.: Linear programming under uncertainty. Management Science 1, 197–206 (1951)
Dhamdhere, K., Ravi, R., Singh, M.: On Two-Stage Stochastic Minimum Spanning Trees. In: Jünger, M., Kaibel, V. (eds.) IPCO 2005. LNCS, vol. 3509, pp. 321–334. Springer, Heidelberg (2005)
Fleischer, L., Koenemann, J., Leonardi, S., Schaefer, G.: Simple cost sharing schemes for multicommodity rent-or-buy and stochastic steiner tree. In: Proceedings of the ACM Symposium on Theory of Computing (STOC), pp. 663–670. ACM Press, New York (2006)
Goemans, M.X., Williamson, D.P.: A General Approximation Technique for Constrained Forest Problems. SIAM Journal on Computing 24(2), 296–317 (1995)
Gupta, A., Pál, M.: Stochastic Steiner Trees Without a Root. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 1051–1063. Springer, Heidelberg (2005)
Gupta, A., Pál, M., Ravi, R., Sinha, A.: What About Wednesday? Approximation Algorithms for Multistage Stochastic Optimization. In: Proceedings of the International Workshop on Approximation and Randomized Algorithms (APPROX-RANDOM), pp. 86–98 (2005)
Gupta, A., Pál, M., Ravi, R., Sinha, A.: Boosted sampling: approximation algorithms for stochastic optimization. In: Gupta, A. (ed.) Proceedings of the ACM Symposium on Theory of Computing (STOC), pp. 417–426. ACM Press, New York (2004)
Gupta, A., Ravi, R., Sinha, A.: An Edge in Time Saves Nine: LP Rounding Approximation Algorithms for Stochastic Network Design. In: Proceedings of the IEEE Symposium on Foundations of Computer Science (FOCS), pp. 218–227. IEEE Computer Society Press, Los Alamitos (2004)
Immorlica, N., Karger, D.R., Minkoff, M., Mirrokni, V.S.: On the costs and benefits of procrastination: approximation algorithms for stochastic combinatorial optimization problems. In: Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 691–700. ACM Press, New York (2004)
Jaillet, P.: Probabilistic Traveling Salesman Problems. PhD thesis (1985)
Murat, C., Paschos, V.Th.: The probabilistic longest path problem. Networks 33(3), 207–219 (1999)
Murat, C., Paschos, V.Th.: A priori optimization for the probabilistic maximum independent set problem. Theoretical Computer Science 270(1–2), 561–590 (2002)
Murat, C., Paschos, V.Th.: On the probabilistic minimum coloring and minimum k-coloring. Discrete Applied Mathematics 154(3), 564–586 (2006)
Vazirani, V.: Approximation Algorithms. LNCS. Springer, Heidelberg (2003)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Paschos, V.T., Telelis, O.A., Zissimopoulos, V. (2007). Steiner Forests on Stochastic Metric Graphs. In: Dress, A., Xu, Y., Zhu, B. (eds) Combinatorial Optimization and Applications. COCOA 2007. Lecture Notes in Computer Science, vol 4616. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73556-4_14
Download citation
DOI: https://doi.org/10.1007/978-3-540-73556-4_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73555-7
Online ISBN: 978-3-540-73556-4
eBook Packages: Computer ScienceComputer Science (R0)