Abstract
We present new combinatorial approximation algorithms for the k-set cover problem. Previous approaches are based on extending the greedy algorithm by efficiently handling small sets. The new algorithms further extend these approaches by utilizing the natural idea of computing large packings of elements into sets of large size. Our results improve the previously best approximation bounds for the k-set cover problem for all values of k≥6. The analysis technique used could be of independent interest; the upper bound on the approximation factor is obtained by bounding the objective value of a factor-revealing linear program.
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Chvátal, V.: A greedy heuristic for the set-covering problem. Math. Oper. Res. 4, 233–235 (1979)
Duh, R., Fürer, M.: Approximation of k-set cover by semi local optimization. In: Proceedings of the 29th Annual ACM Symposium on Theory of Computing (STOC ’97), pp. 256–264 (1997)
Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45(4), 634–652 (1998)
Goldschmidt, O., Hochbaum, D., Yu, G.: A modified greedy heuristic for the set covering problem with improved worst case bound. Inf. Process. Lett. 48, 305–310 (1993)
Halldórsson, M.M.: Approximating discrete collections via local improvements. In: Proceedings of the 6th Annual ACM/SIAM Symposium on Discrete Algorithms (SODA ’95), pp. 160–169 (1995)
Halldórsson, M.M.: Approximating k-set cover and complementary graph coloring. In: Proceedings of the 5th Conference on Integer Programming and Combinatorial Optimization (IPCO ’96). LNCS, vol. 1084, pp. 118–131. Springer, Berlin (1996)
Hazan, E., Safra, S., Schwartz, O.: On the complexity of approximating k-set packing. Comput. Complex. 15(1), 20–39 (2006)
Hurkens, C.A.J., Schrijver, A.: On the size of systems of sets every t of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems. SIAM J. Discrete Math. 2(1), 68–72 (1989)
Jain, K., Mahdian, M., Markakis, E., Saberi, A., Vazirani, V.V.: Greedy facility location algorithms analyzed using dual fitting with factor-revealing LP. J. ACM 50(6), 795–824 (2003)
Johnson, D.S.: Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9, 256–278 (1974)
Kann, V.: Maximum bounded 3-dimensional matching is MAX SNP-complete. Inf. Process. Lett. 37, 27–35 (1991)
Levin, A.: Approximating the unweighted k-set cover problem: greedy meets local search. In: Proceedings of the 4th International Workshop on Approximation and Online Algorithms (WAOA ’06). LNCS, vol. 4368, pp. 290–310. Springer, Berlin (2006)
Lovász, L.: On the ratio of optimal integral and fractional covers. Discrete Math. 13, 383–390 (1975)
Slavík, P.: A tight analysis of the greedy algorithm for set cover. J. Algorithms 25, 237–254 (1997)
Trevisan, L.: Non-approximability results for optimization problems on bounded degree instances. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing (STOC ’01), pp. 453–461 (2001)
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A preliminary version of the results in this paper appeared in Proceedings of the 16th International Symposium on Fundamentals of Computation Theory (FCT ’07), LNCS 4639, Springer, pp. 52–63, 2007. This work was partially supported by the European Union under IST FET Integrated Project 015964 AEOLUS and by the General Secretariat for Research and Technology of the Greek Ministry of Development under programme PENED 2003.
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Athanassopoulos, S., Caragiannis, I. & Kaklamanis, C. Analysis of Approximation Algorithms for k-Set Cover Using Factor-Revealing Linear Programs. Theory Comput Syst 45, 555–576 (2009). https://doi.org/10.1007/s00224-008-9112-3
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DOI: https://doi.org/10.1007/s00224-008-9112-3