Abstract
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?”
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References
Agrawal, S., Ding, Y., Saberi, A., Ye, Y.: Correlation Robust Stochastic Optimization. In: SODA (2010)
Alon, N., Awerbuch, B., Azar, Y., Buchbinder, N., Naor, S.: The Online Set Cover Problem. In: STOC, pp. 100–105 (2003)
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)
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)
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)
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)
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)
Gandhi, R., Khuller, S., Srinivasan, A.: Approximation algorithms for partial covering problems. J. Algorithms 53(1), 55–84 (2004)
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)
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)
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)
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)
Gupta, A., Nagarajan, V., Ravi, R.: Thresholded covering algorithms for robust and max-min optimization (2009), (full version) http://arxiv.org/abs/0912.1045
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)
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)
Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995)
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)
Räcke, H.: Optimal hierarchical decompositions for congestion minimization in networks. In: STOC, pp. 255–264 (2008)
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)
Shmoys, D., Swamy, C.: Stochastic Optimization is (almost) as Easy as Deterministic Optimization. In: FOCS, pp. 228–237 (2004)
Slavík, P.: Improved performance of the greedy algorithm for partial cover. Inf. Process. Lett. 64(5), 251–254 (1997)
Srinivasan, A.: Improved approximation guarantees for packing and covering integer programs. SIAM J. Comput. 29(2), 648–670 (1999)
Sviridenko, M.: A note on maximizing a submodular set function subject to knapsack constraint. Operations Research Letters 32, 41–43 (2004)
Swamy, C.: Algorithms for Probabilistically-Constrained Models of Risk-Averse Stochastic Optimization with Black-Box Distributions (2008), http://arxiv.org/abs/0805.0389
Vondrák, J.: Optimal approximation for the submodular welfare problem in the value oracle model. In: STOC, pp. 67–74 (2008)
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Gupta, A., Nagarajan, V., Ravi, R. (2010). Thresholded Covering Algorithms for Robust and Max-min Optimization. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds) Automata, Languages and Programming. ICALP 2010. Lecture Notes in Computer Science, vol 6198. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14165-2_23
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DOI: https://doi.org/10.1007/978-3-642-14165-2_23
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