Abstract
We study two-stage robust variants of combinatorial optimization problems like Steiner tree, Steiner forest, and uncapacitated facility location. The robust optimization problems, previously studied by Dhamdhere et al. [1], Golovin et al. [6], and Feige et al. [4], are two-stage planning problems in which the requirements are revealed after some decisions are taken in stage one. One has to then complete the solution, at a higher cost, to meet the given requirements. In the robust Steiner tree problem, for example, one buys some edges in stage one after which some terminals are revealed. In the second stage, one has to buy more edges, at a higher cost, to complete the stage one solution to build a Steiner tree on these terminals. The objective is to minimize the total cost under the worst-case scenario. In this paper, we focus on the case of exponentially many scenarios given implicitly. A scenario consists of any subset of k terminals (for Steiner tree), or any subset of k terminal-pairs (for Steiner forest), or any subset of k clients (for facility location). We present the first constant-factor approximation algorithms for the robust Steiner tree and robust uncapacitated facility location problems. For the robust Steiner forest problem with uniform inflation, we present an O(logn)-approximation and show that the problem with two inflation factors is impossible to approximate within O(log1/2 − ε n) factor, for any constant ε> 0, unless NP has randomized quasi-polynomial time algorithms. Finally, we show APX-hardness of the robust min-cut problem (even with singleton-set scenarios), resolving an open question by [1] and [6].
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Dhamdhere, K., Goyal, V., Ravi, R., Singh, M.: How to pay, come what may: Approximation Algorithms for Demand-Robust Covering Problems. In: Proc. of 46th IEEE FOCS (2005)
Dantzig, G.B.: Linear programming under uncertainty. Management Sci. 1, 197–206 (1955)
Fakcharoenphol, J., Rao, S., Talwar, K.: A tight bound on approximating arbitrary metrics by tree metrics. J. Comput. Syst. Sci. 69(3), 485–497 (2004)
Feige, U., Jain, K., Mahdian, M., Mirrokni, V.: Robust Combinatorial Optimization with Exponential Scenarios. In: Fischetti, M., Williamson, D.P. (eds.) IPCO 2007. LNCS, vol. 4513, pp. 439–453. Springer, Heidelberg (2007)
Dahlhaus, E., Johnson, D., Papadimitriou, C., Seymour, P., Yannakakis, M.: The Complexity of Multiterminal Cuts. SIAM J. Comput. 23(4), 864–894 (1994)
Golovin, D., Goyal, V., Ravi, R.: Pay Today for a Rainy Day: Improved Approximation Algorithms for Demand-Robust Min-Cut and Shortest Path Problems. In: Proc. of STACS, pp. 206–217 (2006)
Gupta, A., Pál, M., Ravi, R., Sinha, A.: Boosted sampling: approximation algorithms for stochastic optimization. In: Proc. of 36th ACM STOC (2004)
Gupta, A., Ravi, R., Sinha, A.: An edge in time Saves nine: LP Rounding Approximation Algorithms for Stochastic Network Design. In: Proc. of 45th IEEE FOCS (2004)
Immorlica, N., Karger, D., Minkoff, M., Mirrokni, V.: On the costs and benefits of procrastination: approximation algorithms for stochastic combinatorial optimization problems. In: Proc. of SODA 2004 (2004)
Milnor, J.W.: Games against nature. In: Thrall, R.M., Coomb, C.H., Davis, R.L. (eds.) Decision Processes. Wiley, Chichester
Ravi, R., Sinha, A.: Hedging uncertainty: Approximation algorithms for stochastic optimization problems. In: Bienstock, D., Nemhauser, G.L. (eds.) IPCO 2004. LNCS, vol. 3064, pp. 101–115. Springer, Heidelberg (2004)
Robins, G., Zelikovsky, A.: Improved Steiner tree approximation in graphs. In: Proc. of SODA 2000, pp. 770–779 (2000)
Shmoys, D., Swamy, C.: Stochastic optimization is (almost) as easy as deterministic optimization. In: Proc. of 45th IEEE FOCS 2004 (2004)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Khandekar, R., Kortsarz, G., Mirrokni, V., Salavatipour, M.R. (2008). Two-Stage Robust Network Design with Exponential Scenarios. In: Halperin, D., Mehlhorn, K. (eds) Algorithms - ESA 2008. ESA 2008. Lecture Notes in Computer Science, vol 5193. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-87744-8_49
Download citation
DOI: https://doi.org/10.1007/978-3-540-87744-8_49
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-87743-1
Online ISBN: 978-3-540-87744-8
eBook Packages: Computer ScienceComputer Science (R0)