# Solving Steiner Tree Problems in Graphs with Lagrangian Relaxation

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## 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

- Y.P. Aneja, “An integer linear programming approach to the Steiner problem in graphs,”
*Networks*, vol. 10, pp. 167-178, 1980.Google Scholar - 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 - 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 - 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 - J. Beasley, “An sst-based algorithm for the Steiner problem in graphs,”
*Networks*, vol. 19, pp. 1-16, 1989.Google Scholar - J. Beasley, “An algorithm for the Steiner problem in graphs,”
*Networks*, vol. 14, pp. 147-159, 1984.Google Scholar - 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 - 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 - 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 - 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 - 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
- G.B. Dantzig and P. Wolfe, “Decomposition principle for linear programs,”
*Operations Research*, vol. 8, pp. 101-111, 1960.Google Scholar - C.W. Duin and S. Voß, “Efficient path and vertex exchange in Steiner tree algorithms,”
*Networks*, vol. 29, pp. 89-105, 1997.Google Scholar - 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 - M.X. Goemans and Y.S. Myung, “A catalog of Steiner tree formulations.”
*Networks*, vol. 23, pp. 19-28, 1993.Google Scholar - M. Held, P. Wolfe, and H.P. Crowder, “Validation of subgradient optimization,”
*Mathematical Programming*, vol. 6, pp. 62-88, 1974.Google Scholar - J.B. Hiriart-Urruty and C. Lemaréchal,
*Convex Analysis and Minimization Algorithms*, Springer Verlag, 1991.Google Scholar - 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 - 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 - F.K. Hwang and D.S. Richards, “Steiner tree problems,”
*Networks*, vol. 22, pp. 55-89, 1992.Google Scholar - 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 - T. Koch and A. Martin, “Solving Steiner Tree Problems in Graphs to Optimality,”
*Networks*, vol. 32, pp. 207-232, 1998.Google Scholar - C. Lemaréchal, “An extension of Davidon methods to nondifferentiable problems,”
*Mathematical Programming Study*, vol. 3, pp. 95-109, 1975.Google Scholar - T. Lengauer,
*Combinatorial Algorithms for Integrated Circuit Layout*, Wiley, Chichester, 1990.Google Scholar - A. Lucena, “Steiner problem in graphs: Lagrangian relaxation and cutting-planes.”
*COAL Bull.*, vol. 21, pp. 2-7, 1992.Google Scholar - 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 - N. Maculan, “The Steiner Problem in Graphs,”
*Annals of Discrete Mathematics*, vol. 31, pp. 185-212, 1987.Google Scholar - T.L. Magnanti and T. Wong, “Network design and transportation planning: Models and algorithms.”
*Transp. Science*, vol. 18, pp. 1-55, 1984.Google Scholar - V.J. Rayward-Smith and A. Clare, “On finding Steiner vertices,”
*Networks*, vol. 16, pp. 283-294, 1986.Google Scholar - N. Shor,
*Minimization Methods for Non-Differentiable Functions*, Springer-Verlag, Berlin, 1985.Google Scholar - 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 - 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 - S. Voß, “Steiner's problem in graphs: Heuristic methods,”
*Discrete Applied Mathematics*, vol. 40, pp. 45-72, 1992.Google Scholar - P. Winter, “Steiner problems in networks: A survey,”
*Networks*, vol. 17, pp. 129-167, 1987.Google Scholar - 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 - P. Wolfe, “A method of conjugate subgradients for minimizing nondifferentiable functions,”
*Mathematical Programming Study*, vol. 3, pp. 145-173, 1975.Google Scholar - 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

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