Abstract
We study a variant of the maximum coverage problem which we label the maximum coverage problem with group budget constraints (MCG). We are given a collection of sets \({\cal S} = \{S_1, S_2, \ldots, S_m\}\) where each set S i is a subset of a given ground set X. In the maximum coverage problem the goal is to pick k sets from \({\cal S}\) to maximize the cardinality of their union. In the MCG problem \({\cal S}\) is partitioned into groupsG 1, G 2, ..., G ℓ. The goal is to pick k sets from \({\cal S}\) to maximize the cardinality of their union but with the additional restriction that at most one set be picked from each group. We motivate the study of MCG by pointing out a variety of applications. We show that the greedy algorithm gives a 2-approximation algorithm for this problem which is tight in the oracle model. We also obtain a constant factor approximation algorithm for the cost version of the problem. We then use MCG to obtain the first constant factor approximation algorithms for the following problems: (i) multiple depot k-traveling repairmen problem with covering constraints and (ii) orienteering problem with time windows when the number of time windows is a constant.
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References
Ageev, A., Sviridenko, M.: Pipage Rounding: a New Method of Constructing Algorithms with Proven Performance Guarantee. J. of Combinatorial Optimization (to appear)
Arkin, E., Mitchell, J., Narasimhan, G.: Resource-constrained geometric network optimization. In: Proceedings of SoCG (1998)
S. Arora and G. Karakostas. A 2 + ε approximation for the k-MST problem. In Proceedings of SODA, 2000.
Bar-Yehuda, R., Even, G., Sahar, S.: On Approximating a Geometric Prize-Collecting Traveling Salesman Problem with Time Windows. In: Di Battista, G., Zwick, U. (eds.) ESA 2003. LNCS, vol. 2832, pp. 55–66. Springer, Heidelberg (2003)
Blum, A., Chalasani, P., Coppersmith, D., Pulleyblank, B., Raghavan, P., Sudan, M.: The minimum latency problem. In: Proceedings of STOC (1994)
Bansal, N., Blum, A., Chawla, S., Meyerson, A.: Approximation Algorithms for Deadline-TSP and Vehicle Routing with Time-Windows. In: Proc. of STOC (2004)
Blum, A., Chawla, S., Karger, D., Lane, T., Meyerson, A., Minkoff, M.: Approximation Algorithms for Orienteering and Discounted-Reward TSP. In: Proc. of FOCS (2003)
Chaudhuri, K., Godfrey, B., Rao, S., Talwar, K.: Paths, trees and minimum latency tours. In: Proc. of FOCS (2003)
Chekuri, C., Khanna, S.: A PTAS for the Multiple Knapsack Problem. In: Proc. of SODA (2000)
Elkin, M., Kortsarz, G.: Approximation Algorithm for the Directed Telephone Multicast Problem. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, Springer, Heidelberg (2003)
Fakcharoenphol, J., Harrelson, C., Rao, S.: The k-Traveling Repairmen Problem. In: Proceedings of SODA (2003)
Feige, U.: A Threshold of ln n for Approximating Set Cover. Journal of the ACM 45(4), 634–652 (1998)
Garg, N.: A 3-approximation for the minimum tree spanning k vertices. In: Proceedings of FOCS (1996)
Goemans, M., Kleinberg, J.: An improved approximation ratio for the minimum latency problem. In: Proceedings of SODA (1996)
Hochbaum, D.:(ed.) Approximation Algorithms for NP-Hard Problems. PWS Publishing Company, Boston (1996)
Kortsarz, G.: Personal communication (July 2003)
Khuller, S., Moss, A., Naor, J.: The Budgeted Maximum Coverage Problem. Information Processing Letters 70(1), 39–45 (1999)
Srinivasan, A.: Distributions on level-sets with Applications to Approximation Algorithms. In: Proc. of FOCS (2001)
Tsitsikilis, J.: Special Cases of Traveling Salesman Problem and Repairmen Problems with Time Windows. Networks 22, 228–263 (1992)
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Chekuri, C., Kumar, A. (2004). Maximum Coverage Problem with Group Budget Constraints and Applications. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. RANDOM APPROX 2004 2004. Lecture Notes in Computer Science, vol 3122. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27821-4_7
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DOI: https://doi.org/10.1007/978-3-540-27821-4_7
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