Mathematical Programming

, Volume 150, Issue 1, pp 49–78 | Cite as

The two-level diameter constrained spanning tree problem

Full Length Paper Series B

Abstract

In this article, we introduce the two-level diameter constrained spanning tree problem (2-DMSTP), which generalizes the classical DMSTP by considering two sets of nodes with different latency requirements. We first observe that any feasible solution to the 2-DMSTP can be viewed as a DMST that contains a diameter constrained Steiner tree. This observation allows us to prove graph theoretical properties related to the centers of each tree which are then exploited to develop mixed integer programming formulations, valid inequalities, and symmetry breaking constraints. In particular, we propose a novel modeling approach based on a three-dimensional layered graph. In an extensive computational study we show that a branch-and-cut algorithm based on the latter model is highly effective in practice.

Keywords

Networks/graphs: tree algorithms Integer programming: formulations Layered graphs 

Mathematics Subject Classification (2000)

90C11 90C27 90C57 

References

  1. 1.
    Achuthan, N.R., Caccetta, L., Caccetta, P., Geelen, J.F.: Computational methods for the diameter restricted minimum weight spanning tree problem. Australas. J. Comb. 10, 51–71 (1994)MATHMathSciNetGoogle Scholar
  2. 2.
    Botton, Q., Fortz, B., Gouveia, L., Poss, M.: Benders decomposition for the hop-constrained survivable network design problem. INFORMS J. Comput. 25(1), 13–26 (2013)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Cherkassky, B.V., Goldberg, A.V.: On implementing push-relabel method for the maximum flow problem. Algorithmica 19, 390–410 (1994)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Deo, N., Abdalla, A.: Computing a diameter-constrained minimum spanning tree in parallel. In: Bongiovanni, G.C., et al. (eds.) CIAC, volume 1767 of LNCS, pp. 17–31. Springer, Berlin (2000)Google Scholar
  5. 5.
    Gouveia, L., Magnanti, T.L.: Network flow models for designing diameter-constrained minimum-spanning and Steiner trees. Networks 41(3), 159–173 (2003)CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Gouveia, L., Magnanti, T.L., Requejo, C.: A 2-path approach for odd-diameter-constrained minimum spanning and Steiner trees. Networks 44(4), 254–265 (2004)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Gouveia, L., Magnanti, T.L., Requejo, C.: An intersecting tree model for odd-diameter-constrained minimum spanning and Steiner trees. Ann. OR 146(1), 19–39 (2006)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Gouveia, L., Magnanti, T.L., Requejo, C.: Tight models for special cases of the diameter-constrained minimum spanning tree problem. In: Proceedings of the 3rd International Network Optimization Conference, Spa (2007)Google Scholar
  9. 9.
    Gouveia, L., Simonetti, L., Uchoa, E.: Modeling hop-constrained and diameter-constrained minimum spanning tree problems as Steiner tree problems over layered graphs. Math. Program. 128, 123–148 (2011)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Grötschel, M., Monma, C.L., Stoer, M.: Computational results with a cutting plane algorithm for designing communication networks with low-connectivity constraints. Oper. Res. 40(2), 309–330 (1992)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Gruber, M.: Exact and Heuristic Approaches for Solving the Bounded Diameter Minimum Spanning Tree Problem. PhD thesis, Vienna University of Technology, Vienna, Austria (2009)Google Scholar
  12. 12.
    Gruber, M., Raidl, G.R.: A new 0–1 ILP approach for the bounded diameter minimum spanning tree problem. In: Proceedings of the 2nd International Network Optimization Conference, pp. 178–185, Lisbon (2005)Google Scholar
  13. 13.
    Gruber, M., Raidl, G.R., et al.: (Meta-)heuristic separation of jump cuts in a branch &cut approach for the bounded diameter minimum spanning tree problem. In: Maniezzo, V. (ed.) Matheuristics, Volume 10 of Annals of Information Systems, pp. 209–230. Springer, Berlin (2010)Google Scholar
  14. 14.
    Huygens, D., Labbé, M., Mahjoub, A.R., Pesneau, P.: The two-edge connected hop-constrained network design problem: valid inequalities and branch-and-cut. Networks 49, 116–133 (2007)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Kerivin, H., Mahjoub, A.R.: Design of survivable networks: a survey. Networks 46, 1–21 (2005)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Ljubić, I., Weiskircher, R., Pferschy, U., Klau, G., Mutzel, P., Fischetti, M.: An algorithmic framework for the exact solution of the prize-collecting Steiner tree problem. Math. Program. 105, 427–449 (2006)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Mahjoub, A.R., Simonetti, L., Uchoa, E.: Hop-level flow formulation for the survivable network design with hop constraints problem. Networks 61, 171–179 (2013)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Manyem, P., Stallmann, M.F.M.: Some Approximation Results in Multicasting. Technical Report TR-96-03, North Carolina State University at Raleigh, NC, USA (1996)Google Scholar
  19. 19.
    Noronha, T.F., Ribeiro, C.C., Santos, A.C.: Solving diameter-constrained minimum spanning tree problems by constraint programming. Int. Trans. Oper. Res. 17(5), 653–665 (2010)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Salama, H.F.: Multicast Routing for Real-Time Communication on High-Speed Networks. PhD thesis, Department of Electrical and Computer Engineering, North Carolina State University (1996)Google Scholar
  21. 21.
    Salama, H.F., Reeves, D.S., Viniotis, Y.: Delay-Constrained Shared Multicast Trees. Technical Report, Department of Electrical and Comp. Engg., North Carolina State Univ., USA (1996)Google Scholar
  22. 22.
    Santos, A.C., Lucena, A., Ribeiro, C.C.: Solving diameter constrained minimum spanning tree problems in dense graphs. In: Ribeiro, C.C., Martins, S.L. (eds.) WEA, volume 3059 of LNCS, pp. 458–467. Springer, Berlin (2004)Google Scholar
  23. 23.
    Vik, K.H., Halvorsen, P., Griwodz, C.: Multicast tree diameter for dynamic distributed interactive applications. In: INFOCOM, pp. 1597–1605. IEEE (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2014

Authors and Affiliations

  1. 1.Faculdade de CiênçiasUniversidade de LisboaLisbonPortugal
  2. 2.Department of Statistics and Operations ResearchUniversity of ViennaViennaAustria
  3. 3.Institute of Computer Graphics and AlgorithmsVienna University of TechnologyViennaAustria

Personalised recommendations