Skip to main content
SpringerLink
Log in

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

  • Published:
Theory of Computing Systems Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chvátal, V.: A greedy heuristic for the set-covering problem. Math. Oper. Res. 4, 233–235 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  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 ’97), pp. 256–264 (1997)

  3. Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45(4), 634–652 (1998)

    MATH  MathSciNet  Google Scholar 

  4. 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)

    Article  MATH  MathSciNet  Google Scholar 

  5. 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)

  6. 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)

    Google Scholar 

  7. Hazan, E., Safra, S., Schwartz, O.: On the complexity of approximating k-set packing. Comput. Complex. 15(1), 20–39 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  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 J. Discrete Math. 2(1), 68–72 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  9. 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)

    MathSciNet  Google Scholar 

  10. Johnson, D.S.: Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci. 9, 256–278 (1974)

    Article  MATH  Google Scholar 

  11. Kann, V.: Maximum bounded 3-dimensional matching is MAX SNP-complete. Inf. Process. Lett. 37, 27–35 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  12. 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)

    Chapter  Google Scholar 

  13. Lovász, L.: On the ratio of optimal integral and fractional covers. Discrete Math. 13, 383–390 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  14. Slavík, P.: A tight analysis of the greedy algorithm for set cover. J. Algorithms 25, 237–254 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  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 ’01), pp. 453–461 (2001)

Download references

Author information

Authors and Affiliations

  1. Research Academic Computer Technology Institute & Department of Computer Engineering and Informatics, University of Patras, 26500, Rio, Greece

    Stavros Athanassopoulos, Ioannis Caragiannis & Christos Kaklamanis

Authors
  1. Stavros Athanassopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ioannis Caragiannis
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Christos Kaklamanis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ioannis Caragiannis.

Additional information

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.

Rights and permissions

Reprints and permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00224-008-9112-3

Keywords

Search

Navigation