Approximate k-MSTs and k-Steiner Trees via the Primal-Dual Method and Lagrangean Relaxation
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We consider the problem of computing the minimum-cost tree spanning at least k vertices in an undirected graph. Garg  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 , 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.
KeywordsApproximation Algorithm Lagrangean Relaxation Dual Solution Performance Guarantee Root Vertex
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- 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.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
- 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
- 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
- 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.N. Garg. Personal communication, 1999.Google Scholar
- 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.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.S. Rajagopalan and Vijay V. Vazirani. Logarithmic approximation of minimum weight k trees. Unpublished manuscript, 1995.Google Scholar