Stabilized Branch-and-Price for the Rooted Delay-Constrained Steiner Tree Problem

  • Markus Leitner
  • Mario Ruthmair
  • Günther R. Raidl
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6701)


We consider the rooted delay-constrained Steiner tree problem which arises, e.g., in the design of centralized multicasting networks where quality of service constraints are of concern. We present a mixed integer linear programming formulation based on the concept of feasible paths which has already been considered in the literature for the spanning tree variant. Solving its linear relaxation by column generation has, however, been regarded as computationally not competitive. In this work, we study various possibilities to speed-up the solution of our model by stabilization techniques and embed the column generation procedure in a branch-and-price approach in order to compute proven optimal solutions. Computational results show that the best among the resulting stabilized branch-and-price variants outperforms so-far proposed methods.


Column Generation Variable Neighborhood Search Steiner Tree Problem Feasible Path Path Variable 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Achterberg, T.: SCIP: Solving constraint integer programs. Mathematical Programming Computation 1(1), 1–41 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Amor, H.B., Desrosiers, J.: A proximal trust-region algorithm for column generation stabilization. Computers & Operations Research 33, 910–927 (2006)CrossRefzbMATHGoogle Scholar
  3. 3.
    Amor, H.B., Desrosiers, J., Carvalho, J.M.V.: Dual-optimal inequalities for stabilized column generation. Operations Research 54(3), 454–463 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Amor, H.B., Desrosiers, J., Frangioni, A.: On the choice of explicit stabilizing terms in column generation. Discrete Applied Mathematics 157, 1167–1184 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Barnhart, C., Johnson, E.L., Nemhauser, G.L., Savelsbergh, M.W.P., Vance, P.H.: Branch-and-price: Column generation for solving huge integer programs. Operations Research 46, 316–329 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    de Carvalho, J.M.V.: Using extra dual cuts to accelerate convergence in column generation. INFORMS Journal on Computing 17(2), 175–182 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Desaulniers, G., Desrosiers, J., Solomon, M.M. (eds.): Column Generation. Springer, Heidelberg (2005)zbMATHGoogle Scholar
  8. 8.
    du Merle, O., Villeneuve, D., Desrosiers, J., Hansen, P.: Stabilized column generation. Discrete Mathematics 194(1-3), 229–237 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Dumitrescu, I., Boland, N.: Improved preprocessing, labeling and scaling algorithms for the weight-constrained shortest path problem. Networks 42(3), 135–153 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York (1979)zbMATHGoogle Scholar
  11. 11.
    Ghaboosi, N., Haghighat, A.T.: A path relinking approach for delay-constrained least-cost multicast routing problem. In: 19th IEEE International Conference on Tools with Artificial Intelligence, pp. 383–390 (2007)Google Scholar
  12. 12.
    Gouveia, L., Paias, A., Sharma, D.: Modeling and solving the rooted distance-constrained minimum spanning tree problem. Computers & Operations Research 35(2), 600–613 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Gouveia, L., Simonetti, L., Uchoa, E.: Modeling hop-constrained and diameter-constrained minimum spanning tree problems as Steiner tree problems over layered graphs. Mathematical Programming, 1–26 (2010)Google Scholar
  14. 14.
    Kompella, V.P., Pasquale, J.C., Polyzos, G.C.: Multicasting for multimedia applications. In: Eleventh Annual Joint Conference of the IEEE Computer and Communications Societies, INFOCOM 1992, pp. 2078–2085. IEEE, Los Alamitos (1992)Google Scholar
  15. 15.
    Kou, L., Markowsky, G., Berman, L.: A fast algorithm for Steiner trees. Acta Informatica 15(2), 141–145 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Leggieri, V., Haouari, M., Triki, C.: An exact algorithm for the Steiner tree problem with delays. Electronic Notes in Discrete Mathematics 36, 223–230 (2010)CrossRefzbMATHGoogle Scholar
  17. 17.
    Leitner, M., Raidl, G.R.: Strong lower bounds for a survivable network design problem. In: Haouari, M., Mahjoub, A.R. (eds.) ISCO 2010. Electronic Notes in Discrete Mathematics, vol. 36, pp. 295–302. Elsevier, Amsterdam (2010)Google Scholar
  18. 18.
    Leitner, M., Raidl, G.R., Pferschy, U.: Accelerating column generation for a survivable network design problem. In: Scutellà, M.G., et al. (eds.) Proceedings of the International Network Optimization Conference 2009, Pisa, Italy (2009)Google Scholar
  19. 19.
    Leitner, M., Raidl, G.R., Pferschy, U.: Branch-and-price for a survivable network design problem. Tech. Rep. TR 186–1–10–02, Vienna University of Technology, Vienna, Austria (2010), submitted to NetworksGoogle Scholar
  20. 20.
    Leitner, M., Ruthmair, M., Raidl, G.R.: Stabilized column generation for the rooted delay-constrained Steiner tree problem. In: VII ALIO/EURO – Workshop on Applied Combinatorial Optimization, Porto, Portugal (May 2011)Google Scholar
  21. 21.
    Manyem, P., Stallmann, M.: Some approximation results in multicasting. Tech. Rep. TR-96-03, North Carolina State University (1996)Google Scholar
  22. 22.
    Marsten, R.E., Hogan, W.W., Blankenship, J.W.: The BOXSTEP method for large-scale optimization. Operations Research 23(3), 389–405 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Pessoa, A., Uchoa, E., de Aragão, M., Rodrigues, R.: Exact algorithm over an arc-time-indexed formulation for parallel machine scheduling problems. Mathematical Programming Computation 2(3), 259–290 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Qu, R., Xu, Y., Kendall, G.: A variable neighborhood descent search algorithm for delay-constrained least-cost multicast routing. In: Stützle, T. (ed.) LION 3. LNCS, vol. 5851, pp. 15–29. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  25. 25.
    Rousseau, L.M., Gendreau, M., Feillet, D.: Interior point stabilization for column generation. Operations Research Letters 35(5), 660–668 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Ruthmair, M., Raidl, G.R.: A kruskal-based heuristic for the rooted delay-constrained minimum spanning tree problem. In: Moreno-Díaz, R., Pichler, F., Quesada-Arencibia, A. (eds.) EUROCAST 2009. LNCS, vol. 5717, pp. 713–720. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  27. 27.
    Ruthmair, M., Raidl, G.R.: Variable neighborhood search and ant colony optimization for the rooted delay-constrained minimum spanning tree problem. In: Schaefer, R., Cotta, C., Kołodziej, J., Rudolph, G. (eds.) PPSN XI. LNCS, vol. 6239, pp. 391–400. Springer, Heidelberg (2010)Google Scholar
  28. 28.
    Ruthmair, M., Raidl, G.R.: A layered graph model and an adaptive layers framework to solve delay-constrained minimum tree problems. In: Fifteenth Conference on Integer Programming and Combinatorial Optimization (IPCO XV) (2011) (to appear)Google Scholar
  29. 29.
    Skorin-Kapov, N., Kos, M.: The application of Steiner trees to delay constrained multicast routing: a tabu search approach. In: Proceedings of the 7th International Conference on Telecommunications, vol. 2, pp. 443–448 (2003)Google Scholar
  30. 30.
    Skorin-Kapov, N., Kos, M.: A GRASP heuristic for the delay-constrained multicast routing problem. Telecommunication Systems 32(1), 55–69 (2006)CrossRefGoogle Scholar
  31. 31.
    Vanderbeck, F.: Implementing mixed integer column generation. In: Desaulniers et al [7], pp. 331–358Google Scholar
  32. 32.
    Wentges, P.: Weighted Dantzig-Wolfe decomposition for linear mixed-integer programming. International Transactions in Operational Research 4(2), 151–162 (1997)zbMATHGoogle Scholar
  33. 33.
    Xu, Y., Qu, R.: A GRASP approach for the delay-constrained multicast routing problem. In: Proceedings of the 4th Multidisplinary International Scheduling Conference (MISTA4), pp. 93–104 (2009)Google Scholar
  34. 34.
    Xu, Y., Qu, R.: A hybrid scatter search meta-heuristic for delay-constrained multicast routing problems. Applied Intelligence, 1–13 (2010)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Markus Leitner
    • 1
  • Mario Ruthmair
    • 1
  • Günther R. Raidl
    • 1
  1. 1.Institute of Computer Graphics and AlgorithmsVienna University of TechnologyViennaAustria

Personalised recommendations