A primal-dual approximation algorithm for generalized steiner network problems

Abstract

We present the first polynomial-time approximation algorithm for finding a minimum-cost subgraph having at least a specified number of edges in each cut. This class of problems includes, among others, the generalized Steiner network problem, also called the survivable network design problem. Ifk is the maximum cut requirement of the problem, our solution comes within a factor of 2k of optimal. Our algorithm is primal-dual and shows the importance of this technique in designing approximation algorithms.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    A. Agrawal, P. Klein, andR. Ravi: “When trees collide: An approximation algorithm for the generalized Steiner problem in networks”,Proc. 23rd ACM Symp. on Theory of Computing, 134–144 (1991).

  2. [2]

    P. Berman andV. Ramaiyer: “Improved approximations for the Steiner tree problem”,Proc. 3rd ACM-SIAM Symp. on Discrete Algorithms, 325–334 (1992).

  3. [3]

    G. Dobson: “Worst-case analysis of greedy heuristics for integer programming with non-negative data”,Math. of Oper. Res. 7, 515–531 (1982).

    Google Scholar 

  4. [4]

    G. N. Frederickson andJ. Ja'Ja': “Approximation algorithms for several graph augmentation problems”,SIAM J. Comput. 10, 270–283 (1981).

    Google Scholar 

  5. [5]

    H. N. Gabow, M. X. Goemans, andD. P. Williamson: “An Efficient Approximation algorithm for the survivable network design problem”,Proc. Third Conference on Integer Programming and Combinatorial Optimization, 57–74 (1993).

  6. [6]

    M. X. Goemans andD. J. Bertsimas: “Survivable networks, linear programming relaxations and the parsimonious property”,Math. Programming,60, 145–166 (1993).

    Google Scholar 

  7. [7]

    M. X. Goemans, A. V. Goldberg, S. Plotkin, D. B. Shmoys, É. Tardos andD. P. Williamson: “Improved approximation algorithms for network design problems”,Proc. 5th ACM-SIAM Symp. on Discrete Algorithms, 223–232 (1994).

  8. [8]

    M. X. Goemans andD. P. Williamson: “A general approximation technique for constrained forest problems”,Proc. 3rd ACM-SIAM Symp. on Discrete Algorithms, 307–316 (1992). To appear inSIAM J. on Comput.

  9. [9]

    M. Grötschel, C. L. Monma andM. Stoer: “Design of survivable networks”, to appear in theHandbook in Operations Research and Management Science, Eds: Michael Ball, Thomas Magnanti, Clyde Monma, and George Nemhauser (1992).

  10. [10]

    N. G. Hall andD. S. Hochbaum: “A fast approximation algorithm for the multicovering problem”,Disc. Appl. Math. 15, 35–40 (1986).

    Google Scholar 

  11. [11]

    S. Khuller andU. Vishkin: “Biconnectivity approximations and graph carvings”, Technical Report UMIACS-TR-91-132, Univ. of Maryland (September 1991). Also appears inProc. 24th ACM Symp. on Theory of Computing, 759–770 (1992).

  12. [12]

    P. Klein andR. Ravi: “When cycles collapse: A general approximation technique for constrained two-connectivity problems”,Proc. Third Conference on Integer Programming and Combinatorial Optimization, 39–55 (1993).

  13. [13]

    D. Naor, D. Gusfield, andC. Martel: “A fast algorithm for optimally increasing the edge-connectivity”,Proc. 31st Annual Symp. on Foundations of Computer Science, 698–707 (1990).

  14. [14]

    C. H. Papadimitriou andK. Steiglitz:Combinatorial Optimization: Algorithms and Complexity, Englewood Cliffs, NJ: Prentice-Hall (1982).

    Google Scholar 

  15. [15]

    S. Rajagopalan andV. V. Vazirani: “Primal-dual RNC approximation algorithms for (multi)-set (multi)-cover and covering integer programs”,Proc. 34th Annual Symp. on Foundations of Computer Science, 322–331 (1993).

  16. [16]

    H. Saran, V. Vazirani, andN. Young: “A primal-dual approach to approximation algorithms for network Steiner problems”,Proc. of Indo-US workshop on Cooperative Research in Computer Science, Bangalore, India, 166–168 (1992).

  17. [17]

    A. Z. Zelikovsky: “An 11/6-approximation algorithm for the network Steiner problem”,Algorithmica,9, 463–470 (1993).

    Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

Research supported by an NSF Graduate Fellowship, DARPA contracts N00014-91-J-1698 and N00014-92-J-1799, and AT&T Bell Laboratories.

Research supported in part by Air Force contract F49620-92-J-0125 and DARPA contract N00014-92-J-1799.

Part of this work was done while the author was visiting AT&T Bell Laboratories and Bellcore.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Williamson, D.P., Goemans, M.X., Mihail, M. et al. A primal-dual approximation algorithm for generalized steiner network problems. Combinatorica 15, 435–454 (1995). https://doi.org/10.1007/BF01299747

Download citation

Mathematics Subject Classification (1991)

  • 05 C 40
  • 68 Q 25
  • 90 C 10
  • 90 C 35