Thresholded Covering Algorithms for Robust and Max-min Optimization

  • Anupam Gupta
  • Viswanath Nagarajan
  • R. Ravi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6198)


The general problem of robust optimization is this: one of several possible scenarios will appear tomorrow and require coverage, but things are more expensive tomorrow than they are today. What should you anticipatorily buy today, so that the worst-case covering cost (summed over both days) is minimized? We consider the k-robust model [6,15] where the possible scenarios tomorrow are given by all demand-subsets of size k.

We present a simple and intuitive template for k-robust problems. This gives improved approximation algorithms for the k-robust Steiner tree and set cover problems, and the first approximation algorithms for k-robust Steiner forest, minimum-cut and multicut. As a by-product of our techniques, we also get approximation algorithms for k-max-min problems of the form: “given a covering problem instance, which k of the elements are costliest to cover?


Steiner Tree Covering Problem Online Algorithm Dual Solution Stage Solution 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agrawal, S., Ding, Y., Saberi, A., Ye, Y.: Correlation Robust Stochastic Optimization. In: SODA (2010)Google Scholar
  2. 2.
    Alon, N., Awerbuch, B., Azar, Y., Buchbinder, N., Naor, S.: The Online Set Cover Problem. In: STOC, pp. 100–105 (2003)Google Scholar
  3. 3.
    Calinescu, G., Chekuri, C., Pál, M., Vondrák, J.: Maximizing a monotone submodular function under a matroid constraint. In: Fischetti, M., Williamson, D.P. (eds.) IPCO 2007. LNCS, vol. 4513, pp. 182–196. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  4. 4.
    Dhamdhere, K., Goyal, V., Ravi, R., Singh, M.: How to pay, come what may: Approximation algorithms for demand-robust covering problems. In: FOCS, pp. 367–378 (2005)Google Scholar
  5. 5.
    Fakcharoenphol, J., Rao, S., Talwar, K.: A tight bound on approximating arbitrary metrics by tree metrics. J. Comput. System Sci. 69(3), 485–497 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Feige, U., Jain, K., Mahdian, M., Mirrokni, V.S.: Robust combinatorial optimization with exponential scenarios. In: Fischetti, M., Williamson, D.P. (eds.) IPCO 2007. LNCS, vol. 4513, pp. 439–453. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  7. 7.
    Fisher, M.L., Nemhauser, G.L., Wolsey, L.A.: An analysis of approximations for maximizing submodular set functions II. Mathematical Programming Study 8, 73–87 (1978)MathSciNetGoogle Scholar
  8. 8.
    Gandhi, R., Khuller, S., Srinivasan, A.: Approximation algorithms for partial covering problems. J. Algorithms 53(1), 55–84 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Garg, N.: Saving an epsilon: a 2-approximation for the k-mst problem in graphs. In: STOC 2005: Proceedings of the thirty-seventh annual ACM symposium on Theory of computing, pp. 396–402 (2005)Google Scholar
  10. 10.
    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
  11. 11.
    Golovin, D., Nagarajan, V., Singh, M.: Approximating the k-multicut problem. In: SODA 2006: Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm, pp. 621–630 (2006)Google Scholar
  12. 12.
    Gupta, A., Hajiaghayi, M.T., Nagarajan, V., Ravi, R.: Dial a ride from k-forest. In: Proceedings of the 15th Annual European Symposium on Algorithms, pp. 241–252 (2007)Google Scholar
  13. 13.
    Gupta, A., Nagarajan, V., Ravi, R.: Thresholded covering algorithms for robust and max-min optimization (2009), (full version)
  14. 14.
    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, pp. 691–700 (2004)Google Scholar
  15. 15.
    Khandekar, R., Kortsarz, G., Mirrokni, V.S., Salavatipour, M.R.: Two-stage robust network design with exponential scenarios. In: Halperin, D., Mehlhorn, K. (eds.) Esa 2008. LNCS, vol. 5193, pp. 589–600. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  16. 16.
    Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995)zbMATHGoogle Scholar
  17. 17.
    Nemhauser, G.L., Wolsey, L.A., Fisher, M.L.: An analysis of approximations for maximizing submodular set functions I. Mathematical Programming 14, 265–294 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Räcke, H.: Optimal hierarchical decompositions for congestion minimization in networks. In: STOC, pp. 255–264 (2008)Google Scholar
  19. 19.
    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
  20. 20.
    Shmoys, D., Swamy, C.: Stochastic Optimization is (almost) as Easy as Deterministic Optimization. In: FOCS, pp. 228–237 (2004)Google Scholar
  21. 21.
    Slavík, P.: Improved performance of the greedy algorithm for partial cover. Inf. Process. Lett. 64(5), 251–254 (1997)CrossRefGoogle Scholar
  22. 22.
    Srinivasan, A.: Improved approximation guarantees for packing and covering integer programs. SIAM J. Comput. 29(2), 648–670 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Sviridenko, M.: A note on maximizing a submodular set function subject to knapsack constraint. Operations Research Letters 32, 41–43 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Swamy, C.: Algorithms for Probabilistically-Constrained Models of Risk-Averse Stochastic Optimization with Black-Box Distributions (2008),
  25. 25.
    Vondrák, J.: Optimal approximation for the submodular welfare problem in the value oracle model. In: STOC, pp. 67–74 (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Anupam Gupta
    • 1
  • Viswanath Nagarajan
    • 2
  • R. Ravi
    • 3
  1. 1.Computer Science DepartmentCarnegie Mellon UniversityPittsburghUSA
  2. 2.IBM T.J. Watson Research CenterUSA
  3. 3.Tepper School of BusinessCarnegie Mellon UniversityPittsburghUSA

Personalised recommendations