Analysis of Approximation Algorithms for k-Set Cover Using Factor-Revealing Linear Programs

  • Stavros Athanassopoulos
  • Ioannis Caragiannis
  • Christos Kaklamanis
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4639)

Abstract

We present new combinatorial approximation algorithms for k-set cover. Previous approaches are based on extending the greedy algorithm by efficiently handling small sets. The new algorithms further extend them 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 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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Chvátal, V.: A greedy hueristic for the set-covering problem. Mathematics of Operations Research 4, 233–235 (1979)MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    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 1997), pp. 256–264. ACM Press, New York (1997)CrossRefGoogle Scholar
  3. 3.
    Feige, U.: A threshold of ln n for approximating set cover. Journal of the ACM 45(4), 634–652 (1998)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Goldschmidt, O., Hochbaum, D., Yu, G.: A modified greedy heuristic for the set covering problem with improved worst case bound. Information Processing Letters 48, 305–310 (1993)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Halldórsson, M.M.: Approximating discrete collections via local improvements. In: Proceedings of the 6th Annual ACM/SIAM Symposium on Discrete Algorithms (SODA 1995), pp. 160–169 (1995)Google Scholar
  6. 6.
    Halldórsson, M.M.: Approximating k-set cover and complementary graph coloring. In: Cunningham, W.H., Queyranne, M., McCormick, S.T. (eds.) Integer Programming and Combinatorial Optimization. LNCS, vol. 1084, pp. 118–131. Springer, Heidelberg (1996)Google Scholar
  7. 7.
    Hazan, E., Safra, S., Schwartz, O.: On the complexity of approximating k-set packing. Computational Complexity 15(1), 20–39 (2006)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    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 Journal on Discrete Mathematics 2(1), 68–72 (1989)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Jain, K., Mahdian, M., Markakis, E., Saberi, A., Vazirani, V.V.: Greedy facility location algorithms analyzed using dual fitting with factor-revealing LP. Journal of the ACM 50(6), 795–824 (2003)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Johnson, D.S.: Approximation algorithms for combinatorial problems. Journal of Computer and System Sciences 9, 256–278 (1974)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Kann, V.: Maximum bounded 3-dimensional matching is MAX SNP-complete. Information Processing Letters 37, 27–35 (1991)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Levin, A.: Approximating the unweighted k-set cover problem: greedy meets local search. In: Erlebach, T., Kaklamanis, C. (eds.) WAOA 2006. LNCS, vol. 4368, pp. 290–310. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  13. 13.
    Lovász, L.: On the ratio of optimal integral and fractional covers. Discrete Mathematics 13, 383–390 (1975)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Slavík, P.: A tight analysis of the greedy algorithm for set cover. Journal of Algorithms 25, 237–254 (1997)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    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 2001), pp. 453–461. ACM Press, New York (2001)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Stavros Athanassopoulos
    • 1
  • Ioannis Caragiannis
    • 1
  • Christos Kaklamanis
    • 1
  1. 1.Research Academic Computer Technology Institute &, Department of Computer Engineering and Informatics, University of Patras, 26500 RioGreece

Personalised recommendations