Natural Computing

, Volume 9, Issue 1, pp 29–46 | Cite as

Discrete Particle Swarm Optimization for the minimum labelling Steiner tree problem

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


Particle Swarm Optimization is a population-based method inspired by the social behaviour of individuals inside swarms in nature. Solutions of the problem are modelled as members of the swarm which fly in the solution space. The improvement of the swarm is obtained from the continuous movement of the particles that constitute the swarm submitted to the effect of inertia and the attraction of the members who lead the swarm. This work focuses on a recent Discrete Particle Swarm Optimization for combinatorial optimization, called Jumping Particle Swarm Optimization. Its effectiveness is illustrated on the minimum labelling Steiner tree problem: given an undirected labelled connected graph, the aim is to find a spanning tree covering a given subset of nodes, whose edges have the smallest number of distinct labels.


Combinatorial optimization Discrete Particle Swarm Optimization Heuristics Minimum labelling Steiner tree problem Graphs and networks 



Sergio Consoli was supported by an E.U. Marie Curie Fellowship for Early Stage Researcher Training (EST-FP6) under grant number MEST-CT-2004-006724 at Brunel University (project NET-ACE). The research of José Andrés Moreno-Pérez was partially supported by the projects TIN2005-08404-C04-03 of the Spanish Government (with financial support from the European Union under the FEDER project) and PI042005/044 of the Canary Government.


  1. Al-kazemi B, Mohan CK (2002) Multi-phase discrete particle swarm optimization. In: Fourth international workshop on frontiers in evolutionary algorithms, Kinsale, IrelandGoogle Scholar
  2. Castro-Gutiérrez JP, Landa-Silva D, Moreno-Pérez JA (2008) Exploring feasible and infeasible regions in the vehicle routing problem with time windows using a multi-objective particle swarm optimization approach. In: Proceedings of international workshop on nature inspired cooperatives strategies for optimization (NICSO 2008)Google Scholar
  3. Cerulli R, Fink A, Gentili M, Voß S (2005) Metaheuristics comparison for the minimum labelling spanning tree problem. In: Golden BL, Raghavan S, Wasil EA (eds) The next wave on computing, optimization, and decision technologies. Springer-Verlag, New York, pp 93–106CrossRefGoogle Scholar
  4. Cerulli R, Fink A, Gentili M, Voß S (2006) Extensions of the minimum labelling spanning tree problem. J Telecommun Inf Technol 4:39–45Google Scholar
  5. Chang RS, Leu SJ (1997) The minimum labelling spanning trees. Inf Process Lett 63(5):277–282CrossRefMathSciNetGoogle Scholar
  6. Consoli S, Darby-Dowman K, Mladenović N, Moreno-Pérez JA (2008a) Greedy randomized adaptive search and variable neighbourhood search for the minimum labelling spanning tree problem. Eur J Oper Res 196(2):440–449CrossRefGoogle Scholar
  7. Consoli S, Moreno-Pérez JA, Darby-Dowman K, Mladenović N (2008b) Discrete particle swarm optimization for the minimum labelling Steiner tree problem. In: Krasnogor N, Nicosia G, Pavone M, Pelta D (eds) Nature inspired cooperative strategies for optimization, studies in computational intelligence, vol 129. Springer-Verlag, New York, pp 313–322CrossRefGoogle Scholar
  8. Correa ES, Freitas AA, Johnson CG (2006) A new discrete particle swarm algorithm applied to attribute selection in a bioinformatic data set. In: Proceedings of GECCO 2006, pp 35–42Google Scholar
  9. Demśar J (2006) Statistical comparison of classifiers over multiple data sets. J Mach Learn Res 7:1–30MathSciNetGoogle Scholar
  10. Duin C, Voß S (1999) The Pilot Method: a strategy for heuristic repetition with applications to the Steiner problem in graphs. Networks 34(3):181–191zbMATHCrossRefMathSciNetGoogle Scholar
  11. Friedman M (1940) A comparison of alternative tests of significance for the problem of m rankings. Ann Math Stat 11:86–92zbMATHCrossRefGoogle Scholar
  12. Garey MR, Graham RL, Johnson DS (1977) The complexity of computing Steiner minimal trees. SIAM J Appl Math 32:835–859zbMATHCrossRefMathSciNetGoogle Scholar
  13. Holland JH (1992) Adaptation in natural and artificial systems: an introductory analysis with applications to biology, control, and artificial intelligence. The MIT Press, CambridgeGoogle Scholar
  14. Hollander M, Wolfe DA (1999) Nonparametric statistical methods, 2nd edn. Wiley, New YorkzbMATHGoogle Scholar
  15. Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Proceedings of the 4th IEEE international conference on neural networks, Perth, Australia, pp 1942–1948Google Scholar
  16. 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 4104–4108Google Scholar
  17. Kennedy J, Eberhart R (2001) Swarm intelligence. Morgan Kaufmann Publishers, San FranciscoGoogle Scholar
  18. Krumke SO, Wirth HC (1998) On the minimum label spanning tree problem. Inf Process Lett 66(2):81–85zbMATHCrossRefMathSciNetGoogle Scholar
  19. Martí R (2003) Multi-start methods. In: Glover F, Kochenberger G (eds) Handbook in metaheuristics. Kluwer, Dordrecht, pp 335–368Google Scholar
  20. Martínez-García FJ, Moreno-Pérez JA (2008) Jumping frogs optimization: a new swarm method for discrete optimization. Technical Report DEIOC 3/2008, Department of Statistics, O.R. and Computing, University of La Laguna, Tenerife, SpainGoogle Scholar
  21. Moraglio A, Di Chio C, Togelius J, Poli R (2008) Geometric particle swarm optimization. J Artif Evol Appl. doi: 10.1155/2008/143624
  22. Moreno-Pérez JA, Castro-Gutiérrez JP, Martínez-García FJ, Melián B, Moreno-Vega JM, Ramos J (2007) Discrete Particle Swarm Optimization for the p-median problem. In: Proceedings of the 7th metaheuristics international conference, Montréal, CanadaGoogle Scholar
  23. Nemenyi PB (1963) Distribution-free multiple comparisons. Ph.D. thesis, Princeton University, New JerseyGoogle Scholar
  24. Pampara G, Franken N, Engelbrecht AP (2005) Combining Particle Swarm Optimisation with angle modulation to solve binary problems. In: Proceedings of the IEEE congress on evolutionary computing, vol 1, pp 89–96Google Scholar
  25. Pugh J, Martinoli A (2006) Discrete multi-valued particle swarm optimization. In: Proceedings of IEEE swarm intelligence symposium, vol 1, pp 103–110Google Scholar
  26. Qu R, Xu Y, Castro-Gutiérrez JP, Landa-Silva D (2009) Particle swarm optimization for the Steiner tree in graph and delay-constrained multicast routing problems. Swarm Intelligence (submitted)Google Scholar
  27. Secrest BR (2001) Traveling salesman problem for surveillance mission using Particle Swarm Optimization. Master’s thesis, School of Engineering and Management of the Air Force Institute of TechnologyGoogle Scholar
  28. Tanenbaum AS (1989) Computer Networks. Prentice-Hall, Englewood CliffsGoogle Scholar
  29. Van-Nes R (2002) Design of multimodal transport networks: a hierarchical approach. Delft University Press, DelftGoogle Scholar
  30. Voß S (2000) Modern heuristic search methods for the Steiner tree problem in graphs. In: Du DZ, Smith JM, Rubinstein JH (eds) Advances in Steiner tree. Kluwer, Boston, pp 283–323Google Scholar
  31. Xiong Y, Golden B, Wasil E (2006) Improved heuristics for the minimum labelling spanning tree problem. IEEE Trans Evol Comput 10(6):700–703CrossRefGoogle Scholar
  32. Yang S, Wang M, Jiao L (2004) A Quantum Particle Swarm Optimization. In: Proceedings of CEC2004, the congress on evolutionary computing, vol 1, pp 320–324Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

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

Personalised recommendations