Solving Steiner Tree Problems in Graphs with Lagrangian Relaxation

Journal of Combinatorial Optimization

Volumn 7, Issue 3, 2003, Pages 259-282

  Bahiense, Laura ^a   Barahona, Francisco ^b   Porto, Oscar ^c  

^a FEDERAL UNIVERSITY OF RIO DE JANEIRO   (Brazil)

^b IBM T J WATSON RESEARCH CENTER   (United States)

^c PONTIFICAL CATHOLIC UNIVERSITY OF RIO DE JANEIRO   (Brazil)

Author keywords

Lagrangian relaxation; Steiner trees

Indexed keywords

ALGORITHMS; COMPUTATIONAL METHODS; INTEGER PROGRAMMING; LAGRANGE MULTIPLIERS; PROBLEM SOLVING; RELAXATION PROCESSES; TREES (MATHEMATICS);

FLOW FORMULATIONS; MULTI-COMMODITY FLOW;

GRAPH THEORY;

EID: 0344443734     PISSN: 13826905     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1027368621279     Document Type: Article

References (37)

1. Y.P. Aneja, "An integer linear programming approach to the Steiner problem in graphs," Networks, vol. 10, pp. 167-178, 1980.
2. L. Bahiense, N. Maculan, and C. Sagastizábal, "The volume algorithm revisited: Relation with bundle methods," Mathematical Programming, vol. 94, pp. 41-69, 2002.
3. F. Barahona and R. Anbil, "The volume algorithm: Producing primal solutions with a subgradient method," Mathematical Programming, vol. 87, pp. 385-399, 2000.
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.
5. J. Beasley, "An sst-based algorithm for the Steiner problem in graphs," Networks, vol. 19, pp. 1-16, 1989.
6. J. Beasley, "An algorithm for the Steiner problem in graphs," Networks, vol. 14, pp. 147-159, 1984.
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.
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.
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.
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.
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.
12. G.B. Dantzig and P. Wolfe, "Decomposition principle for linear programs," Operations Research, vol. 8, pp. 101-111, 1960.
13. C.W. Duin and S. Voß, "Efficient path and vertex exchange in Steiner tree algorithms," Networks, vol. 29, pp. 89-105, 1997.
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.
15. M.X. Goemans and Y.S. Myung, "A catalog of Steiner tree formulations." Networks, vol. 23, pp. 19-28, 1993.
16. M. Held, P. Wolfe, and H.P. Crowder, "Validation of subgradient optimization," Mathematical Programming, vol. 6, pp. 62-88, 1974.
17. J.B. Hiriart-Urruty and C. Lemaréchal, Convex Analysis and Minimization Algorithms, Springer Verlag, 1991.
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.
19. F.K. Hwang, D.S. Richards, and P. Winter, "The Steiner tree problem," Annals of Discrete Mathematics, vol. 53. North Holland, Amsterdam, 1992.
20. F.K. Hwang and D.S. Richards, "Steiner tree problems," Networks, vol. 22, pp. 55-89, 1992.
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.
22. T. Koch and A. Martin, "Solving Steiner Tree Problems in Graphs to Optimality," Networks, vol. 32, pp. 207-232, 1998.
23. C. Lemaréchal, "An extension of Davidon methods to nondifferentiable problems," Mathematical Programming Study, vol. 3, pp. 95-109, 1975.
24. T. Lengauer, Combinatorial Algorithms for Integrated Circuit Layout, Wiley, Chichester, 1990.
25. A. Lucena, "Steiner problem in graphs: Lagrangian relaxation and cutting-planes." COAL Bull., vol. 21, pp. 2-7, 1992.
26. A. Lucena. and J. Beasley, "A branch-and-cut algorithm for the Steiner problem in graphs," Networks, vol. 31, pp. 39-59, 1998.
27. N. Maculan, "The Steiner Problem in Graphs," Annals of Discrete Mathematics, vol. 31, pp. 185-212, 1987.
28. T.L. Magnanti and T. Wong, "Network design and transportation planning: Models and algorithms." Transp. Science, vol. 18, pp. 1-55, 1984.
29. V.J. Rayward-Smith and A. Clare, "On finding Steiner vertices, " Networks, vol. 16, pp. 283-294, 1986.
30. N. Shor, Minimization Methods for Non-Differentiable Functions, Springer-Verlag, Berlin, 1985.
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.
32. H. Takahashi and A. Matsuyama, "An approximated solution for the Steiner tree problem in graphs," Math. Japonica, vol. 254, pp. 573-577, 1980.
33. S. Voß, "Steiner's problem in graphs: Heuristic methods, " Discrete Applied Mathematics, vol. 40, pp. 45-72, 1992.
34. P. Winter, "Steiner problems in networks: A survey," Networks, vol. 17, pp. 129-167, 1987.
35. P. Winter and J.M. Smith, "Path-distance heuristics for the Steiner problem in undirected networks," Algorithmica, vol. 7, pp. 309-327, 1992.
36. P. Wolfe, "A method of conjugate subgradients for minimizing nondifferentiable functions," Mathematical Programming Study, vol. 3, pp. 145-173, 1975.
37. R.T. Wong, "A dual ascent approach for Steiner tree problems on a directed graph," Mathematical Programming, vol. 28, pp. 271-287, 1984.