Annals of Operations Research

, 172:71 | Cite as

Variable neighbourhood search for the minimum labelling Steiner tree problem

  • Sergio Consoli
  • Kenneth Darby-Dowman
  • Nenad Mladenović
  • José Andrés Moreno-Pérez
Article

Abstract

We present a study on heuristic solution approaches to the minimum labelling Steiner tree problem, an NP-hard graph problem related to the minimum labelling spanning tree problem. Given an undirected labelled connected graph, the aim is to find a spanning tree covering a given subset of nodes of the graph, whose edges have the smallest number of distinct labels. Such a model may be used to represent many real world problems in telecommunications and multimodal transportation networks. Several metaheuristics are proposed and evaluated. The approaches are compared to the widely adopted Pilot Method and it is shown that the Variable Neighbourhood Search that we propose is the most effective metaheuristic for the problem, obtaining high quality solutions in short computational running times.

Keywords

Metaheuristics Combinatorial optimization Minimum labelling Steiner tree problem Variable neighbourhood search Graphs 

References

  1. Aarts, E., Korst, J., & Michiels, W. (2005). Simulated annealing. In E. K. Burke & G. Kendall (Eds.), Search methodologies: introductory tutorials in optimization and decision support techniques (pp. 187–210). Berlin: Springer. Google Scholar
  2. Avis, D., Hertz, A., & Marcotte, O. (2005). Graph theory and combinatorial optimization. New York: Springer. CrossRefGoogle Scholar
  3. Blum, C., & Roli, A. (2003). Metaheuristics in combinatorial optimization: Overview and conceptual comparison. ACM Computing Surveys, 35(3), 268–308. CrossRefGoogle Scholar
  4. Cerulli, R., Fink, A., Gentili, M., & Voß, S. (2005). Metaheuristics comparison for the minimum labelling spanning tree problem. In B. L. Golden, S. Raghavan, & E. A. Wasil (Eds.), The next wave on computing, optimization, and decision technologies (pp. 93–106). New York: Springer. CrossRefGoogle Scholar
  5. Cerulli, R., Fink, A., Gentili, M., & Voß, S. (2006). Extensions of the minimum labelling spanning tree problem. Journal of Telecommunications and Information Technology, 4, 39–45. Google Scholar
  6. Chang, R. S., & Leu, S. J. (1997). The minimum labelling spanning trees. Information Processing Letters, 63(5), 277–282. CrossRefGoogle Scholar
  7. Consoli, S. (2007). Test datasets for the minimum labelling Steiner tree problem. [Online], available at http://people.brunel.ac.uk/~mapgssc/MLSteiner.htm.
  8. Consoli, S., Darby-Dowman, K., Mladenović, N., & Moreno-Pérez, J. A. (2008a). Greedy randomized adaptive search and variable neighbourhood search for the minimum labelling spanning tree problem. European Journal of Operational Research. doi:10.1016/j.ejor.2008.03.014. Google Scholar
  9. Consoli, S., Darby-Dowman, K., Mladenović, N., & Moreno-Pérez, J. A. (2008b). Heuristics based on greedy randomized adaptive search and variable neighbourhood search for the minimum labelling spanning tree problem. Technical Report TR/01/07, Brunel University, West London, UK. Available: http://hdl.handle.net/2438/737.
  10. Demśar, J. (2006). Statistical comparison of classifiers over multiple data sets. Journal of Machine Learning Research, 7, 1–30. Google Scholar
  11. Duin, C., & Voß, S. (1999). The Pilot method: A strategy for heuristic repetition with applications to the Steiner problem in graphs. Networks, 34(3), 181–191. CrossRefGoogle Scholar
  12. Feo, T. A., & Resende, M. G. C. (1989). A probabilistic heuristic for a computationally difficult set covering problem. Operations Research Letters, 8, 67–71. CrossRefGoogle Scholar
  13. Francis, R. L., McGinnis, L. F., & White, J. A. (1992). Facility layout and location: an analytical approach. Englewood Cliffs: Prentice-Hall. Google Scholar
  14. Friedman, M. (1940). A comparison of alternative tests of significance for the problem of m rankings. Annals of Mathematical Statistics, 11, 86–92. CrossRefGoogle Scholar
  15. Garey, M. R., Graham, R. L., & Johnson, D. S. (1977). The complexity of computing Steiner minimal trees. SIAM Journal on Applied Mathematics, 32, 835–859. CrossRefGoogle Scholar
  16. Grimwood, G. R. (1994). The Euclidean Steiner tree problem: simulated annealing and other heuristics. Master’s thesis, Victoria University, Wellington, New Zealand, URL http://www.isor.vuw.ac.nz/~geoff/thesis.html.
  17. Hansen, P., & Mladenović, N. (1997). Variable neighbourhood search. Computers and Operations Research, 24, 1097–1100. CrossRefGoogle Scholar
  18. Hansen, P., & Mladenović, N. (2003). Variable neighbourhood search. In F. Glover & G. A. Kochenberger (Eds.), Handbook of metaheuristics (pp. 145–184). Norwell: Kluwer. Chap 6. Google Scholar
  19. Hollander, M., & Wolfe, D. A. (1999). Nonparametric statistical methods (2nd edn.). New York: Wiley. Google Scholar
  20. Hwang, F. K., Richards, D. S., & Winter, P. (1992). The Steiner tree problem. Amsterdam: North-Holland. Google Scholar
  21. Karp, R. M. (1975). On the computational complexity of combinatorial problems. Networks, 5, 45–68. Google Scholar
  22. Kennedy, J., & Eberhart, R. (1995). Particle swarm optimization. In Proceedings of the 4th IEEE international conference on neural networks, Perth, Australia (pp. 1942–1948). Google Scholar
  23. Kennedy, J., & Eberhart, R. (1997). A discrete binary version of the particle swarm algorithm. In IEEE conference on systems, man, and cybernetics (Vol. 5, pp. 4104–4108). Google Scholar
  24. Kennedy, J., & Eberhart, R. (2001). Swarm intelligence. San Francisco: Morgan Kaufmann. Google Scholar
  25. Kirkpatrick, S., Gelatt, C. D., & Vecchi, M. P. (1983). Optimization by simulated annealing. Science, 220(4598), 671–680. CrossRefGoogle Scholar
  26. Korte, B., Prömel, H. J., & Steger, A. (1990). Steiner trees in VLSI-layout. In B. Korte, L. Lovász, H. J. Prömel, & A. Schrijver (Eds.), Paths, flows, and VLSI-layout (pp. 185–214). Berlin: Springer. Google Scholar
  27. Krarup, J., & Vajda, S. (1997). On Torricelli’s geometrical solution to a problem of Fermat. IMA. Journal of Mathematics Applied in Business and Industry, 8(3), 215–224. Google Scholar
  28. Krumke, S. O., & Wirth, H. C. (1998). On the minimum label spanning tree problem. Information Processing Letters, 66(2), 81–85. CrossRefGoogle Scholar
  29. Miehle, W. (1958). Link-minimization in networks. Operations Research, 6, 232–243. CrossRefGoogle Scholar
  30. Moreno-Pérez, J. A., Castro-Gutiérrez, J. P., Martínez-García, F. J., Melián, B., Moreno-Vega, J. M., & Ramos, J. (2007). Discrete particle swarm optimization for the p-median problem. In Proceedings of the 7th metaheuristics international conference, Montréal, Canada. Google Scholar
  31. Nemenyi, P. B. (1963). Distribution-free multiple comparisons. Ph.D. thesis, Princeton University, New Jersey. Google Scholar
  32. Pacheco, J., Casado, S., & Nuñez, L. (2007). Use of VNS and TS in classification: variable selection and determination of the linear discrimination function coefficients. IMA Journal of Management Mathematics, 18(2), 191–206. CrossRefGoogle Scholar
  33. Pitsoulis, L. S., & Resende, M. G. C. (2002). Greedy randomized adaptive search procedure. In P. Pardalos & M. G. C. Resende (Eds.), Handbook of applied optimization (pp. 168–183). Oxford: Oxford University Press. Google Scholar
  34. Pérez-Pérez, M., Almeida-Rodríguez, F., & Moreno-Vega, J. M. (2007). A hybrid VNS-path relinking for the p-hub median problem. IMA Journal of Management Mathematics, 18(2), 157–171. CrossRefGoogle Scholar
  35. Raghavan, S., & Anandalingam, G. (2003). Telecommunications network design and management. New York: Springer. Google Scholar
  36. Resende, M. G. C., & Ribeiro, C. C. (2003). Greedy randomized adaptive search procedure. In F. Glover & G. Kochenberger (Eds.), Handbook in metaheuristics (pp. 219–249). Dordrecht: Kluwer. Google Scholar
  37. Tanenbaum, A. S. (1989). Computer networks. Englewood Cliffs: Prentice-Hall. Google Scholar
  38. Van-Nes, R. (2002). Design of multimodal transport networks: a hierarchical approach. Delft: Delft University Press. Google Scholar
  39. Voß, S. (2000). Modern heuristic search methods for the Steiner tree problem in graphs. In D. Z. Du, J. M. Smith, & J. H. Rubinstein (Eds.), Advances in Steiner tree (pp. 283–323). Boston: Kluwer. Google Scholar
  40. Voß, S. (2006). Steiner tree problems in telecommunications. In M. Resende & P. Pardalos (Eds.), Handbook of optimization in telecommunications (pp. 459–492). New York: Springer. Chap 18. CrossRefGoogle Scholar
  41. Voß, S., Martello, S., Osman, I. H., & Roucairol, C. (1999). Meta-heuristics. Advanced and trends local search paradigms for optimization. Norwell: Kluwer. Google Scholar
  42. Voß, S., Fink, A., & Duin, C. (2004). Looking ahead with the Pilot method. Annals of Operations Research, 136, 285–302. CrossRefGoogle Scholar
  43. Wan, Y., Chen, G., & Xu, Y. (2002). A note on the minimum label spanning tree. Information Processing Letters, 84, 99–101. CrossRefGoogle Scholar
  44. Winter, P. (1987). Steiner problem in networks: a survey. Networks, 17, 129–167. CrossRefGoogle Scholar
  45. Xiong, Y., Golden, B., & Wasil, E. (2005a). A one-parameter genetic algorithm for the minimum labelling spanning tree problem. IEEE Transactions on Evolutionary Computation, 9(1), 55–60. CrossRefGoogle Scholar
  46. Xiong, Y., Golden, B., & Wasil, E. (2005b). Worst case behavior of the mvca heuristic for the minimum labelling spanning tree problem. Operations Research Letters, 33(1), 77–80. CrossRefGoogle Scholar
  47. Xiong, Y., Golden, B., & Wasil, E. (2006). Improved heuristics for the minimum labelling spanning tree problem. IEEE Transactions on Evolutionary Computation, 10(6), 700–703. CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Sergio Consoli
    • 1
  • Kenneth Darby-Dowman
    • 1
  • Nenad Mladenović
    • 1
  • José Andrés Moreno-Pérez
    • 2
  1. 1.CARISMA and NET-ACE, School of Information Systems, Computing and MathematicsBrunel UniversityUxbridgeUK
  2. 2.Facultad de Matemáticas, DEIOC, IUDRUniversidad de La LagunaSanta Cruz de TenerifeSpain

Personalised recommendations