Models and Algorithms for Robust Network Design with Several Traffic Scenarios

  • Eduardo Álvarez-Miranda
  • Valentina Cacchiani
  • Tim Dorneth
  • Michael Jünger
  • Frauke Liers
  • Andrea Lodi
  • Tiziano Parriani
  • Daniel R. Schmidt
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7422)

Abstract

We consider a robust network design problem: optimum integral capacities need to be installed in a network such that supplies and demands in each of the explicitly known traffic scenarios can be satisfied by a single-commodity flow. In Buchheim et al. (LNCS 6701, 7–17 (2011)), an integer-programming (IP) formulation of polynomial size was given that uses both flow and capacity variables. We introduce an IP formulation that only uses capacity variables and exponentially many, but polynomial time separable constraints. We discuss the advantages of the latter formulation for branch-and-cut implemenations and evaluate preliminary computational results for the root bounds. We define a class of instances that is difficult for IP-based approaches. Finally, we design and implement a heuristic solution approach based on the exploration of large neighborhoods of carefully selected size and evaluate it on the difficult instances. The results are encouraging, with a good understanding of the trade-off between solution quality and neighborhood size.

Keywords

robust network design cut-set inequalities separation large neighborhood search 

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References

  1. 1.
    Altin, A., Amaldi, E., Belotti, P., Pinar, M.C.: Provisioning virtual private networks under traffic uncertainty. Networks 49(1), 100–115 (2007)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Avella, P., Mattia, S., Sassano, A.: Metric inequalities and the network loading problem. Discrete Optimization 4(1), 103–114 (2007)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Ben-Ameur, W., Kerivin, H.: Routing of uncertain demands. Optimization and Engineering 3, 283–313 (2005)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bienstock, D., Chopra, S., Günlük, O., Tsai, C.H.: Minimum cost capacity installation for multicommodity network flows. Math. Program. 81(2), 177–199 (1998)MATHCrossRefGoogle Scholar
  5. 5.
    Buchheim, C., Liers, F., Oswald, M.: Local cuts revisited. Operations Research Letters 36(4), 430–433 (2008)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Buchheim, C., Liers, F., Sanità, L.: An Exact Algorithm for Robust Network Design. In: Pahl, J., Reiners, T., Voß, S. (eds.) INOC 2011. LNCS, vol. 6701, pp. 7–17. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  7. 7.
    Chekuri, C.: Routing and network design with robustness to changing or uncertain traffic demands. SIGACT News 38(3), 106–128 (2007)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Duffield, N.G., Goyal, P., Greenberg, A.G., Mishra, P.P., Ramakrishnan, K.K., van der Merwe, J.E.: A flexible model for resource management in virtual private networks. In: Proceedings of SIGCOMM, vol. 29, pp. 95–108 (1999)Google Scholar
  9. 9.
    Eisenbrand, F., Grandoni, F., Oriolo, G., Skutella, M.: New approaches for virtual private network design. SIAM Journal on Computing, 706–721 (2007)Google Scholar
  10. 10.
    Erlebach, T., Rüegg, M.: Optimal bandwidth reservation in hose-model VPNs with multi-path routing. In: Proceedings of INFOCOM, vol. 4, pp. 2275–2282 (2004)Google Scholar
  11. 11.
    Fingerhut, J.A., Suri, S., Turner, J.S.: Designing least-cost nonblocking broadband networks. Journal of Algorithms 24(2), 287–309 (1997)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Fiorini, S., Oriolo, G., Sanità, L., Theis, D.O.: The VPN problem with concave costs. SIAM Journal on Discrete Mathematics, 1080–1090 (2010)Google Scholar
  13. 13.
    Goldberg, A.V.: An efficient implementation of a scaling minimum-cost flow algorithm. Journal of Algorithms 22(1), 1–29 (1997)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Gupta, A., Kumar, A., Roughgarden, T.: Simpler and better approximation algorithms for network design. In: Proceedings of STOC, pp. 365–372 (2003)Google Scholar
  15. 15.
    Jain, K.: A factor 2 approximation algorithm for the generalized Steiner network problem. Combinatorica 21(1), 39–60 (2001)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Ford Jr., L.R., Fulkerson, D.R.: A simple algorithm for finding maximal network flows and an application to the hitchcock problem. Canadian Journal of Mathematics 9, 210–218 (1957)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Kerivin, H., Mahjoub, A.R.: Design of survivable networks: A survey. Networks, 1–21 (2005)Google Scholar
  18. 18.
    Koster, A.M.C.A., Kutschka, M., Raack, C.: Towards robust network design using integer linear programming techniques. In: NGI, pp. 1–8 (2010)Google Scholar
  19. 19.
    Magnanti, T.L., Raghavan, S.: Strong formulations for network design problems with connectivity requirements. Networks 45, 61–79 (2005)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    McCormick, S.T., Rao, M.R., Rinaldi, G.: Easy and difficult objective functions for max-cut. Math. Program. 94(2-3, Ser. B), 459–466 (2003)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Raack, C., Koster, A.M.C.A., Orlowski, S., Wessäly, R.: On cut-based inequalities for capacitated network design polyhedra. Networks 57(2), 141–156 (2011)MathSciNetMATHGoogle Scholar
  22. 22.
    Sanità, L.: Robust Network Design. Ph.D. Thesis. Università La Sapienza, Roma (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Eduardo Álvarez-Miranda
    • 1
  • Valentina Cacchiani
    • 1
  • Tim Dorneth
    • 2
  • Michael Jünger
    • 2
  • Frauke Liers
    • 3
  • Andrea Lodi
    • 1
  • Tiziano Parriani
    • 1
  • Daniel R. Schmidt
    • 2
  1. 1.DEISUniversity of BolognaBolognaItaly
  2. 2.Institut für InformatikUniversität zu KölnKölnGermany
  3. 3.Department MathematikFriedrich-Alexander Universität Erlangen-NürnbergErlangenGermany

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