Advertisement

Algorithmica

, Volume 55, Issue 1, pp 42–59 | Cite as

An Improved Analysis for a Greedy Remote-Clique Algorithm Using Factor-Revealing LPs

  • Benjamin BirnbaumEmail author
  • Kenneth J. Goldman
Article

Abstract

Given a positive integer k and a complete graph with non-negative edge weights satisfying the triangle inequality, the remote-clique problem is to find a subset of k vertices having a maximum-weight induced subgraph. A greedy algorithm for the problem has been shown to have an approximation ratio of 4, but this analysis was not shown to be tight. In this paper, we use the technique of factor-revealing linear programs to show that the greedy algorithm actually achieves an approximation ratio of 2, which is tight.

Keywords

Approximation algorithms Dispersion Factor-revealing linear programs Remote-clique 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Baur, C., Fekete, S.P.: Approximation of geometric dispersion problems. Algorithmica 30(3), 451–470 (2001) zbMATHMathSciNetGoogle Scholar
  2. 2.
    Castro, M., Liskov, B.: Practical byzantine fault tolerance. In: OSDI ’99: Proceedings of the 3rd Symposium on Operating Systems Design and Implementation, Berkeley, CA, USA, pp. 173–186. USENIX Association (1999) Google Scholar
  3. 3.
    Chandra, B., Halldórsson, M.M.: Approximation algorithms for dispersion problems. J. Algorithms 38(2), 438–465 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Czygrinow, A.: Maximum dispersion problem in dense graphs. Oper. Res. Lett. 27(5), 223–227 (2000) CrossRefMathSciNetGoogle Scholar
  5. 5.
    Feige, U., Peleg, D., Kortsarz, G.: The dense k-subgraph problem. Algorithmica 29(3), 410–421 (2001) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Fekete, S.P., Meijer, H.: Maximum dispersion and geometric maximum weight cliques. Algorithmica 38(3), 501–511 (2003) CrossRefMathSciNetGoogle Scholar
  7. 7.
    Goemans, M., Kleinberg, J.: An improved approximation ratio for the minimum latency problem. Math. Program. 82, 111–124 (1998) MathSciNetGoogle Scholar
  8. 8.
    Halldórsson, M.M., Iwano, K., Katoh, N., Tokuyama, T.: Finding subsets maximizing minimum structures. SIAM J. Discrete Math. 12(3), 342–359 (1999) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hamming, R.W.: Error detecting and error correcting codes. Bell Syst. Tech. J. 26(2), 147–160 (1950) MathSciNetGoogle Scholar
  10. 10.
    Hassin, R., Rubinstein, S., Tamir, A.: Approximation algorithms for maximum dispersion. Oper. Res. Lett. 21(3), 133–137 (1997) zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    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) CrossRefMathSciNetGoogle Scholar
  12. 12.
    Jain, K., Mahdian, M., Saberi, A.: A new greedy approach for facility location problems. In: STOC ’02: Proceedings of the Thiry-Fourth Annual ACM Symposium on Theory of Computing, New York, NY, USA, pp. 731–740. ACM Press, New York (2002) CrossRefGoogle Scholar
  13. 13.
    Kuby, M.J.: Programming models for facility dispersion: the p-dispersion and maxisum dispersion problems. Geograph. Anal. 19(4), 315–329 (1987) CrossRefGoogle Scholar
  14. 14.
    Mahdian, M., Markakis, E., Saberi, A., Vazirani, V.V.: A greedy facility location algorithm analyzed using dual fitting. In: Goemans, M.X., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) RANDOM-APPROX, Lecture Notes in Computer Science, vol. 2129, pp. 127–137. Springer, New York (2001) Google Scholar
  15. 15.
    McEliese, R.J., Rodemich, E.R., Rumsey, H. Jr., Welch, L.R.: New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities. IEEE Trans. Inf. Theory 23(2), 157–166 (1977) CrossRefGoogle Scholar
  16. 16.
    Mehta, A., Saberi, A., Vazirani, U., Vazirani, V.: Adwords and generalized on-line matching. In: FOCS ’05: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, Washington, DC, USA, pp. 264–273. IEEE Computer Society, Los Alamitos (2005) CrossRefGoogle Scholar
  17. 17.
    Ravi, S.S., Rosencrantz, D.J., Tayi, G.K.: Heuristic and special case algorithms for dispersion problems. Oper. Res. 42(2), 299–310 (1994) zbMATHCrossRefGoogle Scholar
  18. 18.
    Rosenkrantz, D.J., Tayi, G.K., Ravi, S.S.: Facility dispersion problems under capacity and cost constraints. J. Comb. Optim. 4(1), 7–33 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Rosenkrantz, D.J., Tayi, G.K., Ravi, S.S.: Obtaining online approximation algorithms for facility dispersion from offline algorithms. Networks 47(4), 206–217 (2006) zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Tamir, A.: Obnoxious facility location on graphs. SIAM J. Discrete Math. 4(4), 550–567 (1991) zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringUniversity of WashingtonSeattleUSA
  2. 2.Department of Computer Science and EngineeringWashington University in St. LouisSt. LouisUSA

Personalised recommendations