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Journal of Combinatorial Optimization

, Volume 7, Issue 3, pp 259–282 | Cite as

Solving Steiner Tree Problems in Graphs with Lagrangian Relaxation

  • Laura Bahiense
  • Francisco Barahona
  • Oscar Porto
Article

Abstract

This paper presents an algorithm to obtain near optimal solutions for the Steiner tree problem in graphs. It is based on a Lagrangian relaxation of a multi-commodity flow formulation of the problem. An extension of the subgradient algorithm, the volume algorithm, has been used to obtain lower bounds and to estimate primal solutions. It was possible to solve several difficult instances from the literature to proven optimality without branching. Computational results are reported for problems drawn from the SteinLib library.

Steiner trees Lagrangian relaxation 

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References

  1. Y.P. Aneja, “An integer linear programming approach to the Steiner problem in graphs,” Networks, vol. 10, pp. 167-178, 1980.Google Scholar
  2. L. Bahiense, N. Maculan, and C. Sagastiz´abal, “The volume algorithm revisited: Relation with bundle methods,” Mathematical Programming, vol. 94, pp. 41-69, 2002.Google Scholar
  3. F. Barahona and R. Anbil, “The volume algorithm: Producing primal solutions with a subgradient method,” Mathematical Programming, vol. 87, pp. 385-399, 2000.Google Scholar
  4. F. Barahona and F. Chudak. “Solving large scale uncapacitated facility location problems.” in P. Pardalos (Ed.), Approximation and Complexity in Numerical Optimization, Kluwer, pp. 48-62, 2000.Google Scholar
  5. J. Beasley, “An sst-based algorithm for the Steiner problem in graphs,” Networks, vol. 19, pp. 1-16, 1989.Google Scholar
  6. J. Beasley, “An algorithm for the Steiner problem in graphs,” Networks, vol. 14, pp. 147-159, 1984.Google Scholar
  7. P.M. Camerini, L. Frata, and F. Maffioli, “On improving relaxation methods by modified gradient techniques,” Mathematical Programming Study, vol. 3, pp. 26-34, 1975.Google Scholar
  8. S. Chopra, E.R. Gorres, and M.R. Rao, “Solving the Steiner tree problem in graphs using branch-and-cut,” ORSA J. Comput, vol. 4, pp. 320-335, 1992.Google Scholar
  9. S. Chopra and M.R. Rao, “The Steiner tree problem I: formulations, compositions and extension of facets,” Mathematical Programming, vol. 64, pp. 209-229, 1994a.Google Scholar
  10. S. Chopra and M.R. Rao, “The Steiner tree problem II: Properties and classes of facets,” Mathematical Programming, vol. 64, pp. 231-246, 1994b.Google Scholar
  11. A. Claus and N. Maculan, “Une nouvelle formulation du Problème de Steiner sur un graphe,” Technical Report 280, Centre de Recerche sur les Transports, Université de Montréal, 1983.Google Scholar
  12. G.B. Dantzig and P. Wolfe, “Decomposition principle for linear programs,” Operations Research, vol. 8, pp. 101-111, 1960.Google Scholar
  13. C.W. Duin and S. Voß, “Efficient path and vertex exchange in Steiner tree algorithms,” Networks, vol. 29, pp. 89-105, 1997.Google Scholar
  14. M.R. Garey and D.S. Johnson, “The Rectilinear Steiner tree problem is NP-complete,” SIAM Journal on Applied Mathematics, vol. 32, pp. 826-834, 1977.Google Scholar
  15. M.X. Goemans and Y.S. Myung, “A catalog of Steiner tree formulations.” Networks, vol. 23, pp. 19-28, 1993.Google Scholar
  16. M. Held, P. Wolfe, and H.P. Crowder, “Validation of subgradient optimization,” Mathematical Programming, vol. 6, pp. 62-88, 1974.Google Scholar
  17. J.B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms, Springer Verlag, 1991.Google Scholar
  18. K. Holmberg and J. Hellstrand, “Solving the uncapacitated network design problem by a Lagrangian heuristic and branch-and-bound,” Operations Research, vol. 46, pp. 247-259, 1998.Google Scholar
  19. F.K. Hwang, D.S. Richards, and P. Winter, “The Steiner tree problem,” Annals of Discrete Mathematics, vol. 53. North Holland, Amsterdam, 1992.Google Scholar
  20. F.K. Hwang and D.S. Richards, “Steiner tree problems,” Networks, vol. 22, pp. 55-89, 1992.Google Scholar
  21. R.M. Karp, “Reducibility among combinatorial problems,” R.E. Miller and J.W. Thatcher (Eds.), Complexity of Computer Computations, Plenum Press, pp. 85-103, 1972.Google Scholar
  22. T. Koch and A. Martin, “Solving Steiner Tree Problems in Graphs to Optimality,” Networks, vol. 32, pp. 207-232, 1998.Google Scholar
  23. C. Lemaréchal, “An extension of Davidon methods to nondifferentiable problems,” Mathematical Programming Study, vol. 3, pp. 95-109, 1975.Google Scholar
  24. T. Lengauer, Combinatorial Algorithms for Integrated Circuit Layout, Wiley, Chichester, 1990.Google Scholar
  25. A. Lucena, “Steiner problem in graphs: Lagrangian relaxation and cutting-planes.” COAL Bull., vol. 21, pp. 2-7, 1992.Google Scholar
  26. A. Lucena. and J. Beasley, “A branch-and-cut algorithm for the Steiner problem in graphs,” Networks, vol. 31, pp. 39-59, 1998.Google Scholar
  27. N. Maculan, “The Steiner Problem in Graphs,” Annals of Discrete Mathematics, vol. 31, pp. 185-212, 1987.Google Scholar
  28. T.L. Magnanti and T. Wong, “Network design and transportation planning: Models and algorithms.” Transp. Science, vol. 18, pp. 1-55, 1984.Google Scholar
  29. V.J. Rayward-Smith and A. Clare, “On finding Steiner vertices,” Networks, vol. 16, pp. 283-294, 1986.Google Scholar
  30. N. Shor, Minimization Methods for Non-Differentiable Functions, Springer-Verlag, Berlin, 1985.Google Scholar
  31. J. Soukup and W.F. Chow, “Set of test problems for the minimum length connection networks,” ACM/SIGMAP Newsletters, vol. 15, pp. 48-51, 1973.Google Scholar
  32. H. Takahashi and A. Matsuyama, “An approximated solution for the Steiner tree problem in graphs,” Math. Japonica, vol. 254, pp. 573-577, 1980.Google Scholar
  33. S. Voß, “Steiner's problem in graphs: Heuristic methods,” Discrete Applied Mathematics, vol. 40, pp. 45-72, 1992.Google Scholar
  34. P. Winter, “Steiner problems in networks: A survey,” Networks, vol. 17, pp. 129-167, 1987.Google Scholar
  35. P.Winter and J.M. Smith, “Path-distance heuristics for the Steiner problem in undirected networks,” Algorithmica, vol. 7, pp. 309-327, 1992.Google Scholar
  36. P. Wolfe, “A method of conjugate subgradients for minimizing nondifferentiable functions,” Mathematical Programming Study, vol. 3, pp. 145-173, 1975.Google Scholar
  37. R.T. Wong, “A dual ascent approach for Steiner tree problems on a directed graph,” Mathematical Programming, vol. 28, pp. 271-287, 1984.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Laura Bahiense
    • 1
  • Francisco Barahona
    • 2
  • Oscar Porto
    • 3
  1. 1.Universidade Federal do Rio de Janeiro, COPPE-Sistemas e ComputaçãoRio de Janeiro, RJBrazil
  2. 2.IBM T. J. Watson Research CenterYorktown HeightsUSA
  3. 3.PUC-Rio, Dept. de Engenharia ElêtricaRua Marquês de São Vicente 225, Predio Cardeal Leme, Sala 401Rio de Janeiro, RJBrazil

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