Algorithmica

, Volume 18, Issue 1, pp 21–43 | Cite as

An approximation algorithm for minimum-cost vertex-connectivity problems

  • R. Ravi
  • D. P. Williamson
Article

Abstract

We present an approximation algorithm for solving graph problems in which a low-cost set of edges must be selected that has certain vertex-connectivity properties. In the survivable network design problem, a valuer ij for each pair of verticesi andj is given, and a minimum-cost set of edges such that there arer ij vertex-disjoint paths between verticesi andj must be found. In the case for whichr ij ∈{0, 1, 2} for alli, j, we can find a solution of cost no more than three times the optimal cost in polynomial time. In the case in whichr ij =k for alli, j, we can find a solution of cost no more than 2H(k) times optimal, where\(\mathcal{H}(n) = 1 + \tfrac{1}{2} + \cdot \cdot \cdot + \tfrac{1}{n}\). No approximation algorithms were previously known for these problems. Our algorithms rely on a primal-dual approach which has recently led to approximation algorithms for many edge-connectivity problems.

Key Words

Approximation algorithm Vertex connectivity Survivable network design Primal-dual method 

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References

  1. [1]
    A. Agrawal, P. Klein, and R. Ravi. When trees collide: An approximation algorithm for the generalized Steiner problem on networks.SIAM Journal on Computing, 24:440–456, 1995.MATHMathSciNetCrossRefGoogle Scholar
  2. [2]
    K. P. Eswaran and R. E. Tarjan. Augmentation problems.SIAM Journal on Computing, 5:653–665, 1976.MATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    M. Goemans, A. Goldberg, S. Plotkin, D. Shmoys, E. Tardos, and D. Williamson. Improved approximation algorithms for network design problems. InProceedings of the 5th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 223–232, 1994.Google Scholar
  4. [4]
    M. X. Goemans and D. P. Williamson. A general approximation technique for constrained forest problems.SIAM Journal on Computing, 24:296–317, 1995.MATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    M. Grötschel, C. L. Monma, and M. Stoer. Computational results with a cutting plane algorithm for designing communication networks with low-connectivity constraints.Operations Research, 40:309–330, 1992.MATHMathSciNetGoogle Scholar
  6. [6]
    F. Harary. The maximum connectivity of a graph.Proceedings of the National Academy of Sciences, USA, 48:1142–1146, 1962.MATHCrossRefGoogle Scholar
  7. [7]
    T. Hsu. On four-connecting a triconnected graph. InProceedings of the 33rd Annual Symposium on Foundations of Computer Science, pages 70–79, 1992.Google Scholar
  8. [8]
    T. Hsu and V. Ramachandran. A linear time algorithm for triconnectivity augmentation. InProceedings of the 32nd Annual Symposium on Foundations of Computer Science, pages 548–559, 1991.Google Scholar
  9. [9]
    T. Jordán. On the optimal vertex-connectivity augmentation.Journal of Combinatorial Theory, Series B, 63:8–20, 1995.MATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    S. Khuller and B. Raghavachari. Improved approximation algorithms for uniform connectivity problems.Journal of Algorithms, 21:434–450, 1996.MATHMathSciNetCrossRefGoogle Scholar
  11. [11]
    S. Khuller and R. Thurimella. Approximation algorithms for graph augmentation.Journal of Algorithms, 14:214–225, 1993.MATHMathSciNetCrossRefGoogle Scholar
  12. [12]
    P. Klein and R. Ravi. When cycles collapse: A general approximation technique for constrained two-connectivity problems. InProceedings of the Third MPS Conference on Integer Programming and Combinatorial Optimization, pages 39–55, 1993. Also appears as Technical Report CS-92-30, Brown University.Google Scholar
  13. [13]
    K. Menger. Zur allgemeinen Kurventheorie.Fundamenta Mathematicae, 10:96–115, 1927.MATHGoogle Scholar
  14. [14]
    M. Mihail, D. Shallcross, N. Dean, and M. Mostrel. A commercial application of survivable network design: ITP/INPLANS CCS Network Topology Analyzer. InProceedings of the 7th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 279–287, 1996.Google Scholar
  15. [15]
    C. L. Monma and D. F. Shallcross. Methods for designing communication networks with certain two-connected survivability constraintsOperations Research, pages 531–541, 1989.Google Scholar
  16. [16]
    C. H. Papadimitriou and K. SteiglitzCombinatorial Optimization: Algorithms and Complexity. Prentice-Hall, Englewood Cliffs, NJ, 1982.MATHGoogle Scholar
  17. [17]
    M. Rauch. Improved data structures for fully dynamic biconnectivity. InProceedings of the 26th Annual ACM Symposium on Theory of Computing, pages 686–695, 1994. Submitted toSIAM Journal on Computing.Google Scholar
  18. [18]
    R Ravi and D. P. Williamson. An approximation algorithm for minimum-cost vertex-connectivity problems. InProceedings of the 6th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 332–341, 1995.Google Scholar
  19. [19]
    K. Steiglitz, P. Weiner, and D. Kleitman. The design of minimal cost survivable networksIEEE Transactions on Circuit Theory, 16:455–460, 1969.MathSciNetGoogle Scholar
  20. [20]
    M. Stoer.Design of Survivable Networks. Lecture Notes in Mathematics, volume 1531. Springer-Verlag, Berlin, 1992.MATHGoogle Scholar
  21. [21]
    T. Watanabe and A. Nakamura. A minimum 3-connectivity augmentation of a graph.Journal of Computer and System Sciences, 46:91–128, 1993.MATHMathSciNetCrossRefGoogle Scholar
  22. [22]
    D. P. Williamson. On the design of approximation algorithms for a class of graph problems. Ph.D. thesis, MIT, Cambridge, MA, September 1993. Also appears as Technical Report MIT/LCS/TR-584.Google Scholar
  23. [23]
    D. P. Williamson and M. X. Goemans. Computational experience with an approximation algorithm on large-scale Euclidean matching instances.INFORMS Journal on Computing, 8:29–40, 1996.MATHCrossRefGoogle Scholar
  24. [24]
    D. P. Williamson., M. X. Goemans, M. Mihail, and V. V. Vazirani. A primal-dual approximation algorithm for generalized Steiner network problems.Combinatorica, 15:435–454, 1995.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1997

Authors and Affiliations

  • R. Ravi
    • 1
  • D. P. Williamson
    • 2
  1. 1.Graduate School of Industrial AdministrationCarnegie Mellon UniversityPittsburghUSA
  2. 2.IBM TJ Watson Research CenterYorktown HeightsUSA

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