Advertisement

Robust Combinatorial Optimization with Exponential Scenarios

  • Uriel Feige
  • Kamal Jain
  • Mohammad Mahdian
  • Vahab Mirrokni
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4513)

Abstract

Following the well-studied two-stage optimization framework for stochastic optimization  [15,8], we study approximation algorithms for robust two-stage optimization problems with an exponential number of scenarios. Prior to this work, Dhamdhere et al. [8] introduced approximation algorithms for two-stage robust optimization problems with explicitly given scenarios. In this paper, we assume the set of possible scenarios is given implicitly, for example by an upper bound on the number of active clients. In two-stage robust optimization, we need to pre-purchase some resources in the first stage before the adversary’s action. In the second stage, after the adversary chooses the clients that need to be covered, we need to complement our solution by purchasing additional resources at an inflated price. The goal is to minimize the cost in the worst-case scenario. We give a general approach for solving such problems using LP rounding. Our approach uncovers an interesting connection between robust optimization and online competitive algorithms. We use this approach, together with known online algorithms, to develop approximation algorithms for several robust covering problems, such as set cover, vertex cover, and edge cover. We also study a simple buy-at-once algorithm that either covers all items in the first stage or does nothing in the first stage and waits to build the complete solution in the second stage. We show that this algorithm gives tight approximation factors for unweighted variants of these covering problems, but performs poorly for general weighted problems.

Keywords

Approximation Algorithm Robust Optimization Online Algorithm Vertex Cover Vertex Cover Problem 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alon, N., Awerbuch, B., Azar, Y., Buchbinder, N., Naor, J.: A general approach to online network optimization problems. In: SODA, pp. 577–586 (2004)Google Scholar
  2. 2.
    Bar-Yehuda, R., Even, S.: A linear time approximation algorithm for the weighted vertex cover problem. Journal of Algorithms 2, 198–203 (1981)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Ben-Tal, A., Goryashko, A., Guslitzer, E., Nemirovski, A.: Adjustable robust solutions of uncertain linear programs. Mathematical Programming 99(2), 351–376 (2004)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bertsimas, D., Sim, M.: The price of robustness. Operation Research 52, 35–53 (2004)CrossRefMathSciNetMATHGoogle Scholar
  5. 5.
    Bertsimas, D., Sim, M.: Robust discrete optimization and network flows. Mathematical Programming Series B 98, 49–71 (2003)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Birge, J., Louveaux, F.: Introduction to stochastic programming. Springer, Berlin (1997)MATHGoogle Scholar
  7. 7.
    Dantzig, G.B.: Linear programming under uncertainty. Management Science 1, 197–206 (1955)MATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Dhamdhere, K., Goyal, V., Ravi, R., Singh, M.: How to pay, come what may: Approximation algorithms for demand-robust covering problems. In: FOCS (2005)Google Scholar
  9. 9.
    Dinur, I., Guruswami, V., Khot, S., Regev, O.: New multilayered pcp and the hardness of hypergraph vertex cover. SIAM Journal of Computing 34(5), 1129–1146 (2005)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Feige, U.: A threshold of ln n for approximating set cover. JACM 45(4), 634–652 (1998)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    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: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 206–217. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  12. 12.
    Gupta, A., Pal, M., Ravi, R., Sinha, A.: Boosted sampling: Approximation algorithms for stochastic optimization. In: STOC, pp. 170–178 (2004)Google Scholar
  13. 13.
    Gupta, A., Ravi, R., Sinha, A.: An edge in time saves nine: Lp rounding approximation algorithms for stochastic network design. In: FOCS, vol. 45 (2004)Google Scholar
  14. 14.
    Hastad, J.: Clique is hard to approximate. In: FOCS, pp. 627–636 (1996)Google Scholar
  15. 15.
    Immorlica, N., Karger, D., Minkoff, M., Mirrokni, V.S.: On the costs and benefits of procrastination: Approximation algorithms for stochastic combinatorial optimization problems. In: SODA (2004)Google Scholar
  16. 16.
    Lund, C., Yannakakis, M.: On the hardness of approximating minimization problems. JACM 41(5), 960–981 (1994)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Nikulin, Y.: Robustness in combinatorial optimization and scheduling theory: An annotated bibliography. Technical Report SOR-91-13, Statistics and Operation Research (2004), http://www.optimization-online.org/DB_FILE/2004/11/995.pdf
  18. 18.
    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)Google Scholar
  19. 19.
    Siegel, A.: Median Bounds and Their Application. J. Algorithms 38(1), 184–236 (2001)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Shmoys, D., Swamy, S.: Stochastic optimization is (almost) as easy as deterministic optimization. In: FOCS (2004)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Uriel Feige
    • 1
  • Kamal Jain
    • 1
  • Mohammad Mahdian
    • 2
  • Vahab Mirrokni
    • 1
  1. 1.Microsoft Research 
  2. 2.Yahoo! Research 

Personalised recommendations