Abstract
We study the design of approximation algorithms for stochastic combinatorial optimization problems. We formulate the problems in the framework of two-stage stochastic optimization, and provide nearly tight approximations. Our problems range from the simple (shortest path, vertex cover, bin packing) to complex (facility location, set cover), and contain representatives with different approximation ratios.
The approximation ratio of the stochastic variant of a typical problem is of the same order of magnitude as its deterministic counterpart. Furthermore, common techniques for designing approximation algorithms such as LP rounding, the primal-dual method, and the greedy algorithm, can be carefully adapted to obtain these results.
This work was supported in part by NSF grant CCR-0105548 and ITR grant CCR-0122581 (The ALADDIN project).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Arora, S., Sudan, M.: Improved low degree testing and its applications. In: Proceedings of the 29th Annual ACM Symposium on Theory of Computing, pp. 485–495 (1997)
Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation: Combinatorial Optimization Problems and their Approximability Properties. Springer, Berlin (1999)
Balinski, M.L.: On finding integer solutions to linear programs. In: Proc. IBM Scientific Computing Symposium on Combinatorial Problems, pp. 225–248 (1966)
Birge, J., Louveaux, F.: Introduction to Stochastic Programming. Springer, Berlin (1997)
Coffman Jr., E., Garey, M., Johnson, D.: Approximation algorithms for bin-packing: a survey. In: Hochbaum, D.S. (ed.) Approximation Algorithms for NP-hard Problems, PWS, Boston (1997)
Cornuéjols, G., Nemhauser, G., Wolsey, L.: The uncapacitated facility location problem. In: Mirchandani, P., Francis, R. (eds.) Discrete Location Theory, pp. 119–171. Wiley, New York (1990)
Dijkstra, E.: A note on two problems in connexion with graphs. Numerische Mathematik 1, 269–271 (1959)
Fernandez de la Vega, W., Lueker, G.S.: Bin packing can be solved within 1 + ε in linear time. Combinatorica 1, 349–355 (1981)
Garg, N., Konjevod, G., Ravi, R.: A polylogarithmic approximation algorithm for the group Steiner tree problem. Journal of Algorithms 37(1), 66–84 (2000)
Guha, S., Khuller, S.: Greedy strikes back: Improved facility location algorithms. In: Proceedings of the 9th ACM-SIAM Symposium on Discrete Algorithms, pp. 649–657 (1998)
Gupta, A., Kleinberg, J., Kumar, A., Rastogi, R., Yener, B.: Provisioning a virtual private network: A network design problem for multicommodity flow. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pp. 389–398 (2001)
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) (to appear)
Halperin, E., Krauthgamer, R.: Polylogarithmic inapproximability. In: Proceedings of the 35rd Annual ACM Symposium on Theory of Computing, pp. 585–594 (2003)
Håstad, J.: Some optimal inapproximability results. In: Proceedings of the 29th Annual ACM Symposium on Theory of Computing, pp. 1–10 (1997)
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, pp. 684–693 (2004)
Jain, K., Vazirani, V.: Primal-dual approximation algorithms for metric facility location and k-median problems. In: Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science, pp. 2–13 (1999)
Johnson, D.: Approximation algorithms for combinatorial problems. Journal of Computer and System Sciences 9, 256–278 (1974)
Karger, D., Minkoff, M.: Building Steiner trees with incomplete global knowledge. In: Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science, pp. 613–623 (2000)
Klein Haneveld, W.K., van der Vlerk, M.H.: Stochastic Programming, Dept. of Econometrics and OR. University of Groningen, Netherlands (2003)
Kong, N., Schaefer, A.: A factor 1 2 approximation algorithm for a class of two-stage stochastic mixed-integer programs. Manuscript. INFORMS Journal of Computing (2003) (submitted to)
Kumar, A., Swamy, C.: Primal-dual algorithms for connected facility location problems. In: Approximation Algorithms for Combinatorial Optimization, pp. 256–270 (2002)
Lin, J.-H., Vitter, J.: ε-approximations with minimum packing constraint violation. In: Proceedings of the 24th Annual ACM Symposium on Theory of Computing, pp. 771–782 (1992)
Louveaux, F., Peeters, D.: A dual-based procedure for stochastic facility location. Operations Research 40, 564–573 (1992)
Mahdian, M.: Personal communication (2003)
Mahdian, M., Ye, Y., Zhang, J.: A 1.52 approximation algorithm for the uncapacitated facility location problem. In: Approximation Algorithms for Combinatorial Optimization, pp. 229–242 (2002)
Möhring, R., Schulz, A., Uetz, M.: Approximation in stochastic scheduling: The power of LP-based priority policies. Journal of the ACM 46(6), 924–942 (1999)
Monien, B., Speckenmeyer, E.: Ramsey numbers and an approximation algorithm for the vertex cover problem. Acta Informatica 22, 115–123 (1985)
Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. Journal of Computer Systems and Sciences 43, 425–440 (1991)
Ravi, R., Salman, F.S.: Approximation algorithms for the traveling purchaser problem and its variants in network design. In: European Symposium on Algorithms, pp. 29–40 (1999)
Schultz, R., Stougie, L., van der Vlerk, M.H.: Two-stage stochastic integer programming: A survey. Statist. Neerlandica 50(3), 404–416 (1996)
Shmoys, D., Tardos, E., Aardal, K.: Approximation algorithms for facility location problems. In: Proceedings of the 29th ACM Symposium on Theory of Computing, pp. 265–274 (1997)
Skutella, M., Uetz, M.: Scheduling precedence-constrained jobs with stochastic processing times on parallel machines. In: Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 589–590 (2001)
Vazirani, V.: Approximation Algorithms. Springer, Berlin (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ravi, R., Sinha, A. (2004). Hedging Uncertainty: Approximation Algorithms for Stochastic Optimization Problems. In: Bienstock, D., Nemhauser, G. (eds) Integer Programming and Combinatorial Optimization. IPCO 2004. Lecture Notes in Computer Science, vol 3064. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-25960-2_8
Download citation
DOI: https://doi.org/10.1007/978-3-540-25960-2_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-22113-5
Online ISBN: 978-3-540-25960-2
eBook Packages: Springer Book Archive