Steiner Tree Problems in Telecommunications

  • Stefan Voß


Connecting a given set of points at minimum cost may be rated as one of the most important problems in telecommunications network design. Related questions may be formulated in metric spaces as well as in graphs. Given a weighted graph, the Steiner tree problem in graphs asks to determine a minimum cost subgraph spanning a set of specified vertices. This problem may be viewed as the combinatorial optimization problem in telecommunications. In this chapter, we survey Steiner problems from a telecommunications perspective with a special emphasis on the problem in graphs.


Steiner tree problems telecommunications network design graphs 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. F. Adelstein, G.G. Richard, and L. Schwiebert. Distributed multicast tree generation with dynamic group membership. Computer Communications, 26:1105–1128, 2003.CrossRefGoogle Scholar
  2. A. Agrawal, P. Klein, and R. Ravi. When trees collide: An approximation algorithm for the generalized Steiner problem in networks. SIAM Journal on Computing, 24:440–456, 1995.MATHMathSciNetCrossRefGoogle Scholar
  3. M. J. Alexander and G. Robins. A new approach to FPGA routing based on multi-weighted graphs. In Proceedings of the International Workshop on Field-Programmable Gate Arrays, 1994.Google Scholar
  4. Y.P. Aneja. An integer linear programming approach to the Steiner problem in graphs. Networks, 10:167–178, 1980.MATHMathSciNetCrossRefGoogle Scholar
  5. S. Arora. Polynomial-time approximation scheme for Euclidean TSP and other geometric problems. In Proceedings of the Symposium on Foundations of Computer Science, pages 2–11, 1996.Google Scholar
  6. B. Awerbuch, Y. Azar, and Y. Bartal. On-line generalized Steiner problem. Theoretical Computer Science, 324:313–324, 2004.MATHMathSciNetGoogle Scholar
  7. P. Bachhiesl, M. Prossegger, G. Paulus, J. Werner, and H. Stögner. Simulation and optimization of the implementation costs for the last mile of fiber optic networks. Networks and Spatial Economics, 3:467–482, 2004.CrossRefGoogle Scholar
  8. L. Bahiense, F. Barahona, and O. Porto. Solving Steiner tree problems in graphs with Lagrangian relaxation. Journal of Combinatorial Optimization, 7:259–282, 2003.MATHMathSciNetCrossRefGoogle Scholar
  9. A. Balakrishnan and N.R. Patel. Problem reduction methods and a tree generation algorithm for the Steiner network problem. Networks, 17:65–85, 1987.MATHMathSciNetCrossRefGoogle Scholar
  10. F. Bauer and A. Varma. Degree-constrained multicasting in point-to-point networks. In Proceedings IEEE INFOCOM’ 95, pages 369–376, 1995.Google Scholar
  11. J. E. Beasley. An SST-based algorithm for the Steiner problem in graphs. Networks, 19:1–16, 1989.MATHMathSciNetCrossRefGoogle Scholar
  12. J.E. Beasley. Or-library: distributing test problems by electronic mail. Journal of the Operational Research Society, 41:1069–1072, 1990.CrossRefGoogle Scholar
  13. P. Berman and V. Ramaiyer. Improved approximations for the Steiner tree problem. Journal of Algorithms, 17:381–408, 1994.MATHMathSciNetCrossRefGoogle Scholar
  14. B. Bollobás, D. Gamarnik, O. Riordan, and B. Sudakov. On the value of a random minimum weigth Steiner tree. Combinatorica, 24:187–207, 2004.MATHMathSciNetCrossRefGoogle Scholar
  15. A. Candia-Vejar and H. Bravo-Azlan. Performance analysis of algorithms for the Steiner problem in directed networks. Electronic Notes in Discrete Mathematics, 18:67–72, 2004.MathSciNetCrossRefGoogle Scholar
  16. S.A. Canuto, M.G.C. Resende, and C.C. Ribeiro. Local search with perturbations for the prize-collecting Steiner tree problem in graphs. Networks, 38:50–58, 2002.MathSciNetCrossRefGoogle Scholar
  17. D. Chakraborty, S.M.S. Zabir, A. Chayabejara, and G. Chakraborty. A distributed routing method for dynamic multicasting. Telecommunication Systems, 25:299–315, 2004.CrossRefGoogle Scholar
  18. M. Charikar, C. Chekuri, T. Cheung, Z. Dai, A. Goel, S. Guha, and M. Li. Approximation algorithms for directed Steiner problems. Journal of Algorithms, 33:73–91, 1999.MATHMathSciNetCrossRefGoogle Scholar
  19. M. Charikar, J. Naor, and B. Schieber. Resource optimization in QoS multicast routing of real-time multimedia. IEEE/ACM Transactions on Networking, 12:340–348, 2004.CrossRefGoogle Scholar
  20. D. S. Chen. Constrained wirelength minimization of a Steiner tree. Technical report, Department of Electrical Engineering and Computer Science, Northwestern University, Evanston, Illinois, 1994.Google Scholar
  21. S. Chopra, E.R. Gorres, and M.R. Rao. Solving the Steiner tree problem on a graph using branch and cut. ORSA Journal on Computing, 4:320–335, 1992.MATHGoogle Scholar
  22. E. A. Choukhmane. Une heuristique pour le probleme de l’arbre de Steiner. R.A.I.R.O. Recherche Operationelle, 12:207–212, 1978.MATHMathSciNetGoogle Scholar
  23. D. Cieslik. Shortest Connectivity. Springer, New York, 2005.MATHGoogle Scholar
  24. A. Claus and D.J. Kleitman. Cost allocation for a spanning tree. Networks, 3:289–304, 1973.MATHMathSciNetCrossRefGoogle Scholar
  25. E. J. Cockayne and Z. A. Melzak. Steiner’s problem for set-terminals. Quarterly Applied Mathematics, 26:213–218, 1968.MATHMathSciNetGoogle Scholar
  26. R. Courant and H. Robbins. What is Mathematics? Oxford University Press, New York, 1941.Google Scholar
  27. M. Dror, M. Haouari, and J. Chaouachi. Generalized spanning trees. European Journal of Operational Research, 120:583–592, 2000.MATHMathSciNetCrossRefGoogle Scholar
  28. D.-Z. Du. An optimization problem on graphs. Discrete Applied Mathematics, 14:101–104, 1986.MATHMathSciNetCrossRefGoogle Scholar
  29. D.-Z. Du and X. Cheng, editors. Steiner Trees in Industries. Kluwer, Boston, 2001.Google Scholar
  30. D.-Z. Du, B. Lu, H. Ngo, and P.M. Pardalos. Steiner tree problems. In C.A. Floudas and P.M. Pardalos, editors, Encyclopedia of Optimization, volume 5, pages 227–290. Kluwer, Dordrecht, 2001.Google Scholar
  31. D.-Z. Du, J. M. Smith, and J. H. Rubinstein, editors. Advances in Steiner Trees. Kluwer, Boston, 2000.MATHGoogle Scholar
  32. C. Duin. Preprocessing the Steiner problem in graphs. In D.-Z. Du, J. M. Smith, and J. H. Rubinstein, editors, Advances in Steiner Trees, pages 175–233. Kluwer, Boston, 2000.Google Scholar
  33. C. W. Duin. Steiner’s Problem in Graphs: Approximation, Reduction, Estimation. PhD thesis, Faculteit der Economische Wetenschappen en Econometrie, Universiteit van Amsterdam, 1993.Google Scholar
  34. C.W. Duin and A. Volgenant. An edge elimination test for the Steiner problem in graphs. Operations Research Letters, 8:79–83, 1989a.MATHMathSciNetCrossRefGoogle Scholar
  35. C.W. Duin and A. Volgenant. Reducing the hierarchical network design problem. European Journal of Operational Research, 39:332–344, 1989b.MATHMathSciNetCrossRefGoogle Scholar
  36. C.W. Duin and A. Volgenant. Reduction tests for the Steiner problem in graphs. Networks, 19:549–567, 1989c.MATHMathSciNetCrossRefGoogle Scholar
  37. C.W. Duin, A. Volgenant, and S. Voß. Solving group Steiner problems as Steiner problems. European Journal of Operational Research, 154:323–329, 2004.MATHMathSciNetCrossRefGoogle Scholar
  38. C.W. Duin and S. Voß. Steiner tree heuristics-a survey. In H. Dyckhoff, U. Derigs, M. Salomon, and H.C. Tijms, editors, Operations Research Proceedings 1993, pages 485–496, Berlin, 1994. Springer.Google Scholar
  39. C.W. Duin and S. Voß. Efficient path and vertex exchange in Steiner tree algorithms. Networks, 29:89–105, 1997.MATHMathSciNetCrossRefGoogle Scholar
  40. C.W. Duin and S. Voß. The pilot method: A strategy for heuristic repetition with application to the Steiner problem in graphs. Networks, 34:181–191, 1999.MATHMathSciNetCrossRefGoogle Scholar
  41. H. Esbensen. Computing near-optimal solutions to the Steiner problem in a graph using a genetic algorithm. Networks, 26:173–185, 1995.MATHCrossRefGoogle Scholar
  42. A. Fink, G. Schneidereit, and S. Voß. Solving general ring network design problems by meta-heuristics. In M. Laguna and J.L. González Velarde, editors, Computing Tools for Modeling, Optimization and Simulation, pages 91–113. Kluwer, 2000.Google Scholar
  43. H.N. Gabow, Z. Galil, T. Spencer, and R.E. Tarjan. Efficient algorithms for finding minimum spanning trees in undirected and directed graphs. Combinatorica, 6:109–122, 1986.MATHMathSciNetCrossRefGoogle Scholar
  44. M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, San Francisco, 1979.MATHGoogle Scholar
  45. B. Gavish. Topological design of computer communication networks. European Journal of Operational Research, 58:149–172, 1992.MATHCrossRefGoogle Scholar
  46. M. Gendreau, M. Labbé, and G. Laporte. Efficient heuristics for the design of ring networks. Telecommunication Systems, 4:177–188, 1995.CrossRefGoogle Scholar
  47. M. Gendreau, J.-F. Larochelle, and B. Sansò. A tabu search heuristic for the Steiner tree problem. Networks, 34:162–172, 1999.MATHMathSciNetCrossRefGoogle Scholar
  48. A. Goel. Algorithms for network routing, multicasting, switching, and design. PhD thesis, Stanford University, Department of Computer Science, 1999.Google Scholar
  49. M. Goemans and D. Williamson. A general approximation technique for constrained forest problems. SIAM Journal on Computing, 24:296–317,1995.MATHMathSciNetCrossRefGoogle Scholar
  50. M. X. Goemans and Y. S. Myung. A catalog of Steiner tree formulations. Networks, 23:19–28, 1993.MATHMathSciNetCrossRefGoogle Scholar
  51. L. Gouveia. Using variable redefinition for computing lower bounds for minimum spanning and Steiner trees with hop constraints. INFORMS Journal on Computing, 10:180–187, 1998.MATHMathSciNetCrossRefGoogle Scholar
  52. L. Gouveia, T.L. Magnanti, and C. Requejo. A 2-path approach for odd-diameter-constrained minimum spanning and Steiner trees. Networks, 44:254–265, 2004.MATHMathSciNetCrossRefGoogle Scholar
  53. P. Guitart and J.M. Basart. A high performance approximate algorithm for the Steiner problem in graphs, pages 280–293. Springer, Berlin, 1998.Google Scholar
  54. A. Gupta and A. Srinivasan. On the covering Steiner problem. In P.K. Pandya and J. Radhakrishnan, editors, FSTTCS 2003, volume 2914 of Lecture Notes in Computer Science, pages 244–251. Springer, Berlin, 2003.Google Scholar
  55. A. Gupta, A. Srinivasan, and E. Tardos. Cost-sharing mechanisms for network design. In K. Jansen et al., editor, APPROX and RANDOM 2004, pages 139–150. Springer, Berlin, 2004.Google Scholar
  56. M. Hanan. On Steiner’s problem with rectilinear distance. SIAM Journal on Applied Mathematics, 14:255–265, 1966.MATHMathSciNetCrossRefGoogle Scholar
  57. F. K. Hwang and D. S. Richards. Steiner tree problems. Networks, 22:55–89, 1992.MATHMathSciNetCrossRefGoogle Scholar
  58. F. K. Hwang, D.S. Richards, and P. Winter. The Steiner Tree Problem. North-Holland, Amsterdam, 1992.MATHGoogle Scholar
  59. E. Ihler, G. Reich, and P. Widmayer. Class Steiner trees and VLSI-design. Discrete Applied Mathematics, 90:179–194, 1999.MathSciNetCrossRefGoogle Scholar
  60. M. Imase and B. Waxman. The dynamic Steiner tree problem. SIAM Journal of Discrete Mathematics, 4:369–384, 1991.MATHMathSciNetCrossRefGoogle Scholar
  61. V. Jarnik and M. Kössler. O minimalnich grafech, obsahujicich n danych bodu. Casopispro Pestovani Matematiky a Fysiky, pages 223–235, 1934.Google Scholar
  62. D.S. Johnson, M. Minkoff, and S. Phillips. The prize collecting Steiner tree problem: Theory and practice. In Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, pages 760–769. SIAM, 2000.Google Scholar
  63. M. Karpinski, I.I. Mandoiu, A. Olshevsky, and A. Zelikovsky. Improved approximation algorithms for the quality of service multicast tree problem. Algorithmica, 42: 109–120, 2005.MATHMathSciNetCrossRefGoogle Scholar
  64. M. Karpinski and A. Zelikovsky. New approximation algorithms for the Steiner tree problems. Journal of Combinatorial Optimization, 1:47–65, 1997. Also in “Electronic Colloquium on Computational Complexity,” TR95-003 (1995).MATHMathSciNetCrossRefGoogle Scholar
  65. B. N. Khoury, P. M. Pardalos, and D. Z. Du. A test problem generator for the Steiner problem in graphs. ACM Transactions on Mathematical Software, 19:509–522, 1993.MATHCrossRefGoogle Scholar
  66. B.N. Khoury and P.M. Pardalos. A heuristic for the Steiner problem in graphs. Computational Optimization and Applications, 6:5–14, 1996.MATHMathSciNetCrossRefGoogle Scholar
  67. J. Kim, M. Cardei, I. Cardei, and X. Jia. A polynomial time approximation scheme for the grade of service Steiner minimum tree problem. Journal of Global Optimization, 24:427–448, 2002.MathSciNetCrossRefGoogle Scholar
  68. G.W. Klau, I. Ljubic, A. Moser, P. Mutzel, P. Neuner, U. Pferschy, G. Raidl, and R. Weiskircher. Combining a memetic algorithm with integer programming to solve the prize-collecting Steiner tree problem. Technical report, Vienna University of Technology, Vienna, 2004.Google Scholar
  69. T. Koch and A. Martin. Solving Steiner tree problems in graphs to optimality. Networks, 32:207–232, 1998.MATHMathSciNetCrossRefGoogle Scholar
  70. T. Koch, A. Martin, and S. Voß. SteinLib: An updated library on Steiner tree problems in graphs. In D.-Z. Du and X. Cheng, editors, Steiner Trees in Industries, pages 285–325. Kluwer, Boston, 2001.Google Scholar
  71. L. Kou, G. Markowsky, and L. Berman. A fast algorithm for Steiner trees. Acta Informatica, 15:141–145, 1981.MATHMathSciNetCrossRefGoogle Scholar
  72. J.B. Kruskal. On the shortest spanning subtree of a graph and the travelling salesman problem. Proc. Amer. Math. Soc., 7:48–50, 1956.MathSciNetCrossRefGoogle Scholar
  73. A. Lucena. Steiner problem in graphs: Lagrangean relaxation and cutting-planes. Bulletin of the Committee on Algorithms, 21:2–7, 1992.Google Scholar
  74. A. Lucena and J.E. Beasley. A branch and cut algorithm for the Steiner problem in graphs. Networks, 31:39–59, 1998.MATHMathSciNetCrossRefGoogle Scholar
  75. A. Lucena and M. G. C. Resende. Strong lower bounds for the prize collecting Steiner problem in graphs. Discrete Applied Mathematics, 141:277–294, 2004.MATHMathSciNetCrossRefGoogle Scholar
  76. N. Maculan, P. Souza, and A. Candia Vejar. An approach for the Steiner problem in directed graphs. Annals of Operations Research, 33:471–480, 1991.MATHMathSciNetCrossRefGoogle Scholar
  77. S.L. Martins, M.G.C. Resende, C.C. Ribeiro, and P.M. Pardalos. A parallel GRASP for the Steiner tree problem in graphs using a hybrid local search strategy. Journal of Global Optimization, 17:267–283, 2000.MATHMathSciNetCrossRefGoogle Scholar
  78. K. Mehlhorn. A faster approximation algorithm for the Steiner problem in graphs. Information Processing Letters, 27:125–128, 1988.MATHMathSciNetCrossRefGoogle Scholar
  79. Z. Miller, D. Pritikin, M. Perkel, and I. H. Sudborough. The sequential sum problem and performance bounds on the greedy algorithm for the on-line Steiner problem. Networks, 45:143–164, 2005.MATHMathSciNetCrossRefGoogle Scholar
  80. M. Minoux. Efficient greedy heuristics for Steiner tree problems using reoptimization and supermodularity. INFOR, 28:221–233, 1990.MATHGoogle Scholar
  81. P. Mirchandani. The multi-tier tree problem. INFORMS Journal on Computing, 8: 202–218, 1996.MATHCrossRefGoogle Scholar
  82. R. Novak, J. Rugelj, and G. Kandus. Steiner tree based distributed multicast routing. In D.-Z. Du and X. Cheng, editors, Steiner Trees in Industries, pages 327–352. Kluwer, Boston, 2001.Google Scholar
  83. C. A. S. Oliveira and P. M. Pardalos. A survey of combinatorial optimization problems in multicast routing. Computers & Operations Research, 32:1953–1981, 2005.MATHCrossRefGoogle Scholar
  84. L.J. Osborne and B.E. Gillett. A comparison of two simulated annealing algorithms applied to the directed Steiner problem on networks. ORSA Journal on Computing, 3:213–225, 1991.MATHGoogle Scholar
  85. J. Pearl. Heuristics: Intelligent Search Techniques for Computer Problem Solving. Addison-Wesley, Reading, 1984.Google Scholar
  86. J. Plesnik. A bound for the Steiner tree problem in graphs. Math. Slovaca, 31:155–163, 1981.MATHMathSciNetGoogle Scholar
  87. J. Plesnik. Worst-case relative performance of heuristics for the Steiner problem in graphs. Acta Math. Univ. Comenianae, 60:269–284, 1991.MATHMathSciNetGoogle Scholar
  88. J. Plesnik. Heuristics for the Steiner problem in graphs. Discrete Applied Mathematics, 37/38:451–463, 1992.MathSciNetCrossRefGoogle Scholar
  89. T. Polzin and S. Vahdati Daneshmand. Algorithmen für das Steiner-Problem. Diploma thesis, University of Dortmund, 1997.Google Scholar
  90. T. Polzin and S. Vahdati Daneshmand. A comparison of Steiner tree relaxations. Discrete Applied Mathematics, 112:241–261, 2001a.MATHMathSciNetCrossRefGoogle Scholar
  91. T. Polzin and S. Vahdati Daneshmand. Improved algorithms for the Steiner problem in networks. Discrete Applied Mathematics, 112:263–300, 2001b.MATHMathSciNetCrossRefGoogle Scholar
  92. R.C. Prim. Shortest connection networks and some generalizations. Bell Syst. Techn. J., 36:1389–1401, 1957.Google Scholar
  93. H. J. Prömel and A. Steger. The Steiner Tree Problem. Vieweg, Wiesbaden, 2002.MATHGoogle Scholar
  94. S. K. Rao, P. Sadayappan, F. K. Hwang, and P. W. Shor. The rectilinear Steiner arborescence problem. Algorithmica, 7:277–288, 1992.MATHMathSciNetCrossRefGoogle Scholar
  95. R. Ravi, M.V. Marathe, S.S. Ravi, D.J. Rosenkrantz, and H.B. Hunt III. Approximation algorithms for degree-constrained minimum-cost network-design problems. Algorithmica, 31:58–78, 2001.MATHMathSciNetCrossRefGoogle Scholar
  96. R. Ravi and A. Sinha. Hedging uncertainty: Approximation algorithms for stochastic optimization problems. In D. Bienstock and G. Nemhauser, editors, IPCO 2004, pages 101–115. Springer, Berlin, 2004.Google Scholar
  97. V. J. Rayward-Smith. The computation of nearly minimal Steiner trees in graphs. Int. J. Math. Educ. Sci. Technol., 14:15–23, 1983.MATHMathSciNetCrossRefGoogle Scholar
  98. V. J. Rayward-Smith and A. Clare. On finding Steiner vertices. Networks, 16:283–294, 1986.MATHMathSciNetCrossRefGoogle Scholar
  99. C.C. Ribeiro and M.C. De Souza. Tabu search for the Steiner problem in graphs. Networks, 36:138–146, 2000.MATHMathSciNetCrossRefGoogle Scholar
  100. C.C. Ribeiro, E. Uchoa, and R.F. Werneck. A hybrid GRASP with perturbations for the Steiner problem in graphs. INFORMS Journal on Computing, 14:228–246, 2002.MathSciNetCrossRefGoogle Scholar
  101. M. B. Richey and R. G. Parker. On multiple Steiner subgraph problems. Networks, 16:423–438, 1986.MATHMathSciNetCrossRefGoogle Scholar
  102. G. Robins and A. Zelikovsky. Improved Steiner tree approximation in graphs. In Proceedings of the Eleventh Annual ACM-SIAM Symposium on Discrete Algorithms, pages 770–779, 2000.Google Scholar
  103. I. Rosseti, M. Poggi de Aragao, C. Ribeiro, E. Uchoa, and R.F. Werneck. New benchmark instances for the Steiner problem in graphs. In M.G.C. Resende and J.P de Sousa, editors, Metaheuristics: Computer Decision-Making, pages 601–614. Kluwer, Boston, 2003.Google Scholar
  104. J.-J. Salazar-González. The Steiner cycle polytope. European Journal of Operational Research, pages 671–679, 2003.Google Scholar
  105. A. Segev. The node-weighted Steiner tree problem. Networks, 17:1–17, 1987.MATHMathSciNetCrossRefGoogle Scholar
  106. M. Servit. Heuristic algorithms for rectilinear Steiner trees. Digital Processes, 7: 21–32, 1981.MATHGoogle Scholar
  107. W.W. Sharkey. Network models in economics. In M.O. Ball, T.L. Magnanti, C.L. Monma, and G.L. Nemhauser, editors, Network Routing, pages 713–765. North-Holland, Amsterdam, 1995.Google Scholar
  108. D. Skorin-Kapov. On cost allocation in Steiner tree networks. In D.-Z. Du and X. Cheng, editors, Steiner Trees in Industries, pages 353–376. Kluwer, Boston, 2001.Google Scholar
  109. L. Sondergeld and S. Voß. A multi-level star-shaped intensification and diversification approach in tabu search for the Steiner tree problem in graphs. Technical report, TU Braunschweig, 1996.Google Scholar
  110. J. Soukup and W.F. Chow. Set of test problems for the minimum length connection networks. ACM/SIGMAP Newsletter, 15:48–51, 1973.Google Scholar
  111. J. Steiner. Aufgaben und Lehrsätze, erstere aufzulösen, letztere zu beweisen. Journal für die reine und angewandte Mathematik, 13:361–364, 1835.MATHCrossRefGoogle Scholar
  112. H. Takahashi and A. Matsuyama. An approximate solution for the Steiner problem in graphs. Math. Japonica, 24:573–577, 1980.MATHMathSciNetGoogle Scholar
  113. E. Uchoa. Local search with perturbations for the prize-collecting Steiner tree problem in graphs. Technical report, Universidade Federal Fluminense, Niterói, Brazil, 2005.Google Scholar
  114. M. G. A. Verhoeven, M. E. M. Severens, and E. H. L. Aarts. Local search for Steiner trees in graphs. In V.J. Rayward-Smith, LH. Osman, CR. Reeves, and G.D. Smith, editors, Modern Heuristic Search Methods, pages 117–129. Wiley, Chichester, 1996.Google Scholar
  115. S. Voß and C.W. Duin. Heuristic methods for the rectilinear Steiner arborescence problem. Engineering Optimization, 21:121–145, 1993.CrossRefGoogle Scholar
  116. S. Voß. Steiner-Probleme in Graphen. Hain, Frankfurt/Main, 1990a.MATHGoogle Scholar
  117. S. Voß. A survey on some generalizations of Steiner’s problem. In B. Papathanassiu and K. Giatas, editors, 1st Balkan Conference on Operational Research Proceedings, pages 41–51. Hellenic Productivity Center, Thessaloniki, 1990b.Google Scholar
  118. S. Voß. Steiner’s problem in graphs: heuristic methods. Discrete Applied Mathematics, 40:45–72, 1992.MATHMathSciNetCrossRefGoogle Scholar
  119. S. Voß. Worst case performance of some heuristics for Steiner’s problem in directed graphs. Information Processing Letters, 48:99–105, 1993.MATHMathSciNetCrossRefGoogle Scholar
  120. S. Voß. The Steiner tree problem with hop constraints. Annals of Operations Research, 86:321–345, 1999.MATHMathSciNetCrossRefGoogle Scholar
  121. S. Voß. Modern heuristic search methods for the Steiner tree problem in graphs. In D.-Z. Du, J. M. Smith, and J. H. Rubinstein, editors, Advances in Steiner Trees, pages 283–323. Kluwer, Boston, 2000.Google Scholar
  122. S. Voß and K. Gutenschwager. A chunking based genetic algorithm for the Steiner tree problem in graphs. In P.M. Pardalos and D.-Z. Du, editors, Network Design: Connectivity and Facilities Location, volume 40 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 335–355. AMS, Princeton, 1998.Google Scholar
  123. D.M. Warme, P. Winter, and M. Zachariasen. Exact algorithms for plane Steiner tree problems: A computational study. In D.-Z. Du, J. M. Smith, and J. H. Rubinstein, editors, Advances in Steiner Trees, pages 81–116. Kluwer, Boston, 2000.Google Scholar
  124. B.M. Waxman and M. Imase. Worst-case performance of Rayward-Smith’s Steiner tree heuristic. Information Processing Letters, 29:283–287, 1988.MATHMathSciNetCrossRefGoogle Scholar
  125. J. F. Weng. Generalized Steiner problem and hexagonal coordinate system (in Chinese). Acta Math. Appl. Sinica, 8:383–397, 1985.MATHMathSciNetGoogle Scholar
  126. J.F. Weng. Steiner trees an curved surfaces. Graphs and Combinatorics, 17:353–363, 2001.MATHMathSciNetCrossRefGoogle Scholar
  127. P. Widmayer. Fast approximation algorithms for Steiner’s problem in graphs. Habilitation thesis, Institut für Angewandte Informatik und formale Beschreibungsverfahren, University Karlsruhe, 1986.Google Scholar
  128. P. Winter. Steiner problem in networks: a survey. Networks, 17:129–167, 1987.MATHMathSciNetCrossRefGoogle Scholar
  129. P. Winter and J. MacGregor Smith. Path-distance heuristics for the Steiner problem in undirected networks. Algorithmica, 7:309–327, 1992.MATHMathSciNetCrossRefGoogle Scholar
  130. R. T. Wong. A dual ascent approach for Steiner tree problems on a directed graph. Mathematical Programming, 28:271–287, 1984.MATHMathSciNetCrossRefGoogle Scholar
  131. K. Woolston and S. Albin. The design of centralized networks with reliability and availability constraints. Computers & Operations Research, 15:207–217, 1988.CrossRefGoogle Scholar
  132. B. Y. Wu and K.-M. Chao. Spanning Trees and Optimization Problems. Chapman & Hall / CRC, Boca Raton, 2004.MATHGoogle Scholar
  133. J. Xu, S.Y. Chiu, and F. Glover. A probabilistic tabu search for the telecommunications network design. Combinatorial Optimization: Theory and Practice, 1:69–94, 1996a.Google Scholar
  134. J. Xu, S.Y. Chiu, and F. Glover. Using tabu search to solve Steiner tree-star problem in telecommunications network design. Telecommunication Systems, 6:117–125, 1996b.CrossRefGoogle Scholar
  135. H.-H. Yen and F.Y.-S. Lin. Near-optimal tree-based access network design. Computer Communications, 28:236–245, 2005.CrossRefGoogle Scholar
  136. M. Zachariasen. The rectilinear Steiner problem: A tutorial. In D.-Z. Du and X. Cheng, editors, Steiner Trees in Industries, pages 467–507. Kluwer, Boston, 2001.Google Scholar
  137. M. Zachariasen and A. Rohe. Rectilinear group Steiner trees and applications in VLSI design. Mathematical Programming, 94:407–433, 2003.MATHMathSciNetCrossRefGoogle Scholar
  138. A. Zelikovsky. A series of approximation algorithms for the acyclic directed Steiner tree problem. Algorithmica, 18:99–110, 1997.MATHMathSciNetCrossRefGoogle Scholar
  139. A. Z. Zelikovsky. An 11/6-approximation algorithm for the network Steiner problem. Algorithmica, 9:463–470, 1993a.MATHMathSciNetCrossRefGoogle Scholar
  140. A.Z. Zelikovsky. A faster approximation algorithm for the Steiner tree problem in graphs. Information Processing Letters, 46:79–83, 1993b.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • Stefan Voß
    • 1
  1. 1.Institute of Information SystemsUniversity of HamburgHamburgGermany

Personalised recommendations