Advertisement

Abstract

We give the first constant factor approximation algorithm for the asymmetric Virtual Private Network (Vpn) problem with arbitrary concave costs. We even show the stronger result, that there is always a tree solution of cost at most 2·OPT and that a tree solution of (expected) cost at most 49.84·OPT can be determined in polynomial time.

For the case of linear cost we obtain a \((2+\varepsilon\frac{\mathcal R}{\mathcal S})\)-approximation algorithm for any fixed ε > 0, where \(\mathcal{S}\) and \(\mathcal{R}\) (\(\mathcal{R} \geq \mathcal{S}\)) denote the outgoing and ingoing demand, respectively.

Furthermore, we answer an outstanding open question about the complexity status of the so called balanced \(\textsc{Vpn}\) problem by proving its NP-hardness.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Fingerhut, J., Suri, S., Turner, J.: Designing least-cost nonblocking broadband networks. Journal of Algorithms 24(2), 287–309 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Gupta, A., Kleinberg, J., Kumar, A., Rastogi, R., Yener, B.: Provisioning a virtual private network: A network design problem for multicommodity flow. In: Proc. of the 33rd ACM Symposium on Theory of Computing (STOC), pp. 389–398 (2001)Google Scholar
  3. 3.
    Erlebach, T., Rüegg, M.: Optimal bandwidth reservation in hose-model vpns with multi-path routing. In: Proc. of the 23rd Annual Joint Conference of the IEEE Computer and Communications Societies INFOCOM, vol. 4, pp. 2275–2282 (2004)Google Scholar
  4. 4.
    Eisenbrand, F., Grandoni, F., Oriolo, G., Skutella, M.: New approaches for virtual private network design. SIAM Journal on Computing 37(3), 706–721 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Italiano, G., Leonardi, S., Oriolo, G.: Design of trees in the hose model: the balanced case. Operation Research Letters 34(6), 601–606 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hurkens, C., Keijsper, J., Stougie, L.: Virtual private network design: A proof of the tree routing conjecture on ring networks. SIAM Journal on Discrete Mathematics 21, 482–503 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Grandoni, F., Kaibel, V., Oriolo, G., Skutella, M.: A short proof of the vpn tree routing conjecture on ring networks. Operation Research Letters 36(3), 361–365 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Goyal, N., Olver, N., Shepherd, B.: The vpn conjecture is true. In: Proc. of the 40th annual ACM symposium on Theory of computing (STOC), pp. 443–450 (2008)Google Scholar
  9. 9.
    Eisenbrand, F., Grandoni, F.: An improved approximation algorithm for virtual private network design. In: Proc. of the 16th ACM-SIAM Symposium on Discrete Algorithms (SODA) (2005)Google Scholar
  10. 10.
    Fiorini, S., Oriolo, G., Sanità, L., Theis, D.: The vpn problem with concave costs (submitted manuscript) (2008)Google Scholar
  11. 11.
    Chekuri, C.: Routing and network design with robustness to changing or uncertain traffic demands. SIGACT News 38(3), 106–128 (2007)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Sanità, L.: Robust Network Design. PhD thesis, Università Sapienza di Roma (2008)Google Scholar
  13. 13.
    Goyal, N., Olver, N., Shepherd, B.: Personal communication (2008)Google Scholar
  14. 14.
    Guha, S., Meyerson, A., Munagala, K.: A constant factor approximation for the single sink edge installation problems. In: Proc. of the 33rd Annual ACM Symposium on Theory of Computing (STOC), New York, pp. 383–388 (2001)Google Scholar
  15. 15.
    Talwar, K.: The single-sink buy-at-bulk LP has constant integrality gap. In: Proc. of the 9th International Conference on Integer programming and combinatorial optimization (IPCO), pp. 475–486 (2002)Google Scholar
  16. 16.
    Gupta, A., Kumar, A., Roughgarden, T.: Simpler and better approximation algorithms for network design. In: Proc. of the 35th Annual ACM Symposium on Theory of Computing (STOC), New York, pp. 365–372 (2003)Google Scholar
  17. 17.
    Jothi, R., Raghavachari, B.: Improved approximation algorithms for the single-sink buy-at-bulk network design problems. In: Hagerup, T., Katajainen, J. (eds.) SWAT 2004. LNCS, vol. 3111, pp. 336–348. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  18. 18.
    Grandoni, F., Italiano, G.F.: Improved approximation for single-sink buy-at-bulk. In: Asano, T. (ed.) ISAAC 2006. LNCS, vol. 4288, pp. 111–120. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  19. 19.
    Cook, W., Cunningham, W., Pulleyblank, W., Schrijver, A.: Combinatorial optimization. John Wiley & Sons Inc., New York (1998)zbMATHGoogle Scholar
  20. 20.
    Schrijver, A.: Combinatorial optimization. Polyhedra and efficiency. vol. A. Algorithms and Combinatorics, vol. 24. Springer, Berlin (2003)Google Scholar
  21. 21.
    van Zuylen, A.: Deterministic sampling algorithms for network design. In: Halperin, D., Mehlhorn, K. (eds.) Esa 2008. LNCS, vol. 5193, pp. 830–841. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  22. 22.
    Eisenbrand, F., Grandoni, F., Rothvoß, T., Schäfer, G.: Approximating connected facility location problems via random facility sampling and core detouring. In: Proc. of the 19th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1174–1183 (2008)Google Scholar
  23. 23.
    Chlebík, M., Chlebíková, J.: Approximation hardness of the steiner tree problem on graphs. In: Penttonen, M., Schmidt, E.M. (eds.) SWAT 2002. LNCS, vol. 2368, p. 170. Springer, Heidelberg (2002)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Thomas Rothvoß
    • 1
  • Laura Sanità
    • 1
  1. 1.Institute of MathematicsEPFLLausanneSwitzerland

Personalised recommendations