Advertisement

Approximate k-MSTs and k-Steiner Trees via the Primal-Dual Method and Lagrangean Relaxation

  • Fabián A. Chudak
  • Tim Roughgarden
  • David P. Williamson
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2081)

Abstract

We consider the problem of computing the minimum-cost tree spanning at least k vertices in an undirected graph. Garg [10] gave two approximation algorithms for this problem. We show that Garg’s al- gorithms can be explained simply with ideas introduced by Jain and Vazirani for the metric uncapacitated facility location and k-median problems [15], in particular via a Lagrangean relaxation technique to- gether with the primal-dual method for approximation algorithms. We also derive a constant-factor approximation algorithm for the k-Steiner tree problem using these ideas, and point out the common features of these problems that allow them to be solved with similar techniques.

Keywords

Approximation Algorithm Lagrangean Relaxation Dual Solution Performance Guarantee Root Vertex 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    S. Arora and G. Karakostas. Approximation schemes for minimum latency problems. In Proceedings of the 31st Annual ACM Symposium on the Theory of Computing, pages 688–693, 1999.Google Scholar
  2. 2.
    S. Arora and G. Karakostas. A 2 + ∈ approximation algorithm for the k-MST problem. In Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 754–759, 2000.Google Scholar
  3. 3.
    S. Arya and H. Ramesh. A 2.5-factor approximation algorithm for the k-MST problem. Information Processing Letters, 65:117–118, 1998.CrossRefMathSciNetGoogle Scholar
  4. 4.
    B. Awerbuch, Y. Azar, A. Blum, and S. Vempala. Improved approximation guarantees for minimum-weight k-trees and prize-collecting salesmen. SIAM Journal on Computing, 28(1):254–262, 1999.CrossRefMathSciNetGoogle Scholar
  5. 5.
    M. Bern and P. Plassmann. The Steiner problem with edge lengths 1 and 2. Information Processing Letters, 32:171–176, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    A. Blum, P. Chalasani, D. Coppersmith, W. Pulleyblank, P. Raghavan, and M. Sudan. The minimum latency problem. In Proceedings of the 26th Annual ACM Symposium on the Theory of Computing, pages 163–171, 1994.Google Scholar
  7. 7.
    A. Blum, R. Ravi, and S. Vempala. A constant-factor approximation algorithm for the k-MST problem. Journal of Computer and System Sciences, 58(1):101–108, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    M. Charikar and S. Guha. Improved combinatorial algorithms for the facility location and k-median problems. In Proceedings of the 40th Annual Symposium on Foundations of Computer Science, pages 378–388, 1999.Google Scholar
  9. 9.
    M. Fischetti, H. Hamacher, K. Jørnsten, and F. Maffioli. Weighted k-cardinality trees: Complexity and polyhedral structure. Networks, 24:11–21,1994.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    N. Garg. A 3-approximation for the minimum tree spanning k vertices. In Proceedings of the 37th Annual Symposium on Foundations of Computer Science, pages 302–309, 1996.Google Scholar
  11. 11.
    N. Garg. Personal communication, 1999.Google Scholar
  12. 12.
    M. Goemans and J. Kleinberg. An improved approximation ratio for the minimum latency problem. Mathematical Programming, 82:111–124, 1998.MathSciNetGoogle Scholar
  13. 13.
    M. X. Goemans and D. P. Williamson. A general approximation technique for constrained forest problems. SIAM Journal on Computing, 24:296–317, 1995.zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    M. X. Goemans and D. P. Williamson. The primal-dual method for approximation algorithms and its application to network design problems. In D. S. Hochbaum, editor, Approximation Algorithms for NP-Hard Problems, chapter 4, pages 144–191. PWS Publishing Company, 1997.Google Scholar
  15. 15.
    K. Jain and V. V. Vazirani. Primal-dual approximation algorithms for metric facility location and k-median problems. In Proceedings of the 40th Annual Symposium on Foundations of Computer Science, pages 2–13, 1999.Google Scholar
  16. 16.
    S. Rajagopalan and Vijay V. Vazirani. Logarithmic approximation of minimum weight k trees. Unpublished manuscript, 1995.Google Scholar
  17. 17.
    R. Ravi, R. Sundaram, M. V. Marathe, D. J. Rosenkrantz, and S. S. Ravi. Spanning trees short or small. SIAM Journal on Discrete Mathematics, 9:178–200, 1996.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Fabián A. Chudak
    • 1
  • Tim Roughgarden
    • 2
  • David P. Williamson
    • 3
  1. 1.Tellabs Research CenterMAUSA
  2. 2.Department of Computer ScienceCornell UniversityUpson Hall, IthacaUSA
  3. 3.IBM Almaden Research CenterSan JoseUSA

Personalised recommendations