Skip to main content
SpringerLink
Log in

Network design through forests with degree- and role-constrained minimum spanning trees

  • Published:
Journal of Heuristics Aims and scope Submit manuscript

Abstract

Finding the degree-constrained minimum spanning tree (DCMST) of a graph is a widely studied NP-hard problem. One of its most important applications is network design. Here we deal with a new variant of the DCMST problem, which consists of finding not only the degree- but also the role-constrained minimum spanning tree (DRCMST), i.e., we add constraints to restrict the role of the nodes in the tree to root, intermediate or leaf node. Furthermore, we do not limit the number of root nodes to one, thereby, generally, building a forest of DRCMSTs. The modeling of network design problems can benefit from the possibility of generating more than one tree and determining the role of the nodes in the network. We propose a novel permutation-based representation to encode these forests. In this new representation, one permutation simultaneously encodes all the trees to be built. We simulate a wide variety of DRCMST problem instances which we optimize using different evolutionary computation algorithms encoding individuals of the population using the proposed representation. To illustrate the applicability of our approach, we formulate the trans-European transport network as a DRCMST problem. In this network design, we simultaneously optimize nine transport corridors and show that it is straightforward using the proposed representation to add constraints depending on the specific characteristics of the network.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Notes

  1. http://sci2s.ugr.es/sicidm/.

  2. http://ec.europa.eu/transport/themes/infrastructure/ten-t-guidelines/corridors/doc/ten-t-corridor-map-2013.

References

  • Aledo, J.A., Gámez, J.A., Molina, D.: Tackling the rank aggregation problem with evolutionary algorithms. Appl. Math. Comput. 222, 632–644 (2013)

    MathSciNet  MATH  Google Scholar 

  • Anton-Sanchez, L., Bielza, C., Larrañaga, P.: Towards optimal neuronal wiring through estimation of distribution algorithms. In: Proceedings of the Fifteenth Annual Conference on Genetic and Evolutionary Computation, GECCO ’13 Companion, pp. 1647–1650 (2013)

  • Bergmann, B., Hommel, G.: Improvements of general multiple test procedures for redundant systems of hypotheses. In: Multiple Hypotheses Testing, Medizinische Informatik und Statistik, vol. 70, pp. 100–115. Springer, Berlin (1988)

  • Bielza, C., Fernández del Pozo, J.A., Larrañaga, P., Bengoetxea, E.: Multidimensional statistical analysis of the parameterization of a genetic algorithm for the optimal ordering of tables. Expert Syst. Appl. 37(1), 804–815 (2010)

    Article  Google Scholar 

  • Ceberio, J., Mendiburu, A., Lozano, J.A.: Introducing the Mallows model on estimation of distribution algorithms. In: Neural Information Processing. Lecture Notes in Computer Science, vol. 7063, pp. 461–470. Springer, Berlin (2011)

  • Ceberio, J., Irurozki, E., Mendiburu, A., Lozano, J.A.: A review on estimation of distribution algorithms in permutation-based combinatorial optimization problems. Prog. Artif. Intell. 1(1), 103–117 (2012)

    Article  MATH  Google Scholar 

  • Ceberio, J., Irurozki, E., Mendiburu, A., Lozano, J.A.: A distance-based ranking model estimation of distribution algorithm for the flowshop scheduling problem. IEEE Trans. Evol. Comput. 18(2), 286–300 (2014)

    Article  Google Scholar 

  • Ceberio, J., Mendiburu, A., Lozano, J.A.: Kernels of mallows models for solving permutation-based problems. In: Proceedings of the Genetic and Evolutionary Computation Conference, GECCO 2015, Madrid, July 11–15, 2015, pp. 505–512 (2015)

  • Cobb, H., Grefenstette, J.: Genetic algorithms for tracking changing environments. In: Proceedings of the Fifth International Conference on Genetic Algorithms, Morgan Kaufmann, pp. 523–530 (1993)

  • Czajko, M.M., Wojciechowski, J.: Tree-based access network design under requirements for an aggregation network. Elektron. Konstr. Technol. Zastos. 4, 23–27 (2009)

    Google Scholar 

  • Delbem, A., de Carvalho, A., Policastro, C., Pinto, A., Honda, K., García, A.: Node-depth encoding for evolutionary algorithms applied to network design. In: Genetic and Evolutionary Computation. Lecture Notes in Computer Science, vol. 3102, pp. 678–687. Springer, Berlin (2004)

  • Delbem, A.C.B., de Lima, T.W., Telles, G.P.: Efficient forest data structure for evolutionary algorithms applied to network design. IEEE Trans. Evolut. Comput. 16(6), 829–846 (2012)

    Article  Google Scholar 

  • Demšar, J.: Statistical comparisons of classifiers over multiple data sets. J. Mach. Learn. Res. 7, 1–30 (2006)

    MathSciNet  MATH  Google Scholar 

  • Derrac, J., García, S., Molina, D., Herrera, F.: A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms. Swarm Evolut. Comput. 1(1), 3–18 (2011)

    Article  Google Scholar 

  • Durillo, J.J., Nebro, A.J.: jMetal: a java framework for multi-objective optimization. Adv. Eng. Softw. 42(10), 760–771 (2011)

    Article  Google Scholar 

  • Friedman, M.: The use of ranks to avoid the assumption of normality implicit in the analysis of variance. J. Am. Stat. Assoc. 32(200), 675–701 (1937)

    Article  MATH  Google Scholar 

  • Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co, New York (1979)

    MATH  Google Scholar 

  • Holland, J.H.: Adaptation in Natural and Artificial Systems: An Introductory Analysis with Applications to Biology, Control, and Artificial Intelligence. University of Michigan Press, Ann Arbor (1975)

    MATH  Google Scholar 

  • Irurozki, E., Calvo, B., Lozano, J.A.: Sampling and Learning the Mallows and Generalized Mallows Models Under the Cayley Distance. University of the Basque Country, Spain (2014)

    Google Scholar 

  • Knowles, J., Corne, D., Oates, M.: A new evolutionary approach to the degree constrained minimum spanning tree problem. IEEE Trans. Evol. Comput. 4, 125–134 (2000)

    Article  Google Scholar 

  • Krishnamoorthy, M., Ernst, A., Sharaiha, Y.: Comparison of algorithms for the degree constrained minimum spanning tree. J. Heuristics 7(6), 587–611 (2001)

    Article  MATH  Google Scholar 

  • Kruskal, J.B.: On the shortest spanning subtree of a graph and the traveling salesman problem. Proc. Am. Math. Soc. 7(1), 48–50 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  • Larrañaga, P., Kuijpers, C.M.H., Murga, R.H., Inza, I., Dizdarevic, S.: Genetic algorithms for the travelling salesman problem: a review of representations and operators. Artif. Intell. Rev. 13(2), 129–170 (1999)

    Article  Google Scholar 

  • Larrañaga, P., Lozano, J.A. (eds.): Estimation of Distribution Algorithms: A New Tool for Evolutionary Computation. Kluwer, Boston (2002)

    MATH  Google Scholar 

  • Mallows, C.L.: Non-null ranking models. Biometrika 44(1–2), 114–130 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  • Prim, R.C.: Shortest connection networks and some generalizations. Bell Syst. Technol. J. 36, 1389–1401 (1957)

    Article  Google Scholar 

  • Raidl, G., Julstrom, B.: Edge sets: an effective evolutionary coding of spanning trees. IEEE Trans. Evol. Comput. 7(3), 225–239 (2003)

    Article  Google Scholar 

  • Reeves, C.R.: A genetic algorithm for flowshop sequencing. Comput. Oper. Res. 22(1), 5–13 (1995)

    Article  MATH  Google Scholar 

  • Ruiz, R., Maroto, C.: A comprehensive review and evaluation of permutation flowshop heuristics. Eur. J. Oper. Res. 165(2), 479–494 (2005)

    Article  MATH  Google Scholar 

  • Sedgewick, R., Wayne, K.: Algorithms, 4th edn. Addison-Wesley, Boston (2011)

    Google Scholar 

  • Soak, S.M., Corne, D., Ahn, B.H.: A new encoding for the degree constrained minimum spanning tree problem. In: Knowledge-Based Intelligent Information and Engineering Systems. Lecture Notes in Computer Science, vol. 3213, pp. 952–958. Springer, Berlin (2004)

  • Syswerda, G.: A study of reproduction in generational and steady-state genetic algorithms. Found. Genet. Algorithms 1, 94–101 (1991)

    Article  Google Scholar 

  • Tsutsui, S.: Node histogram vs. edge histogram: a comparison of probabilistic model-building genetic algorithms in permutation domains. In: IEEE Congress on Evolutionary Computation, 2006, pp. 1939–1946 (2006)

Download references

Acknowledgements

This work has been partially supported by the Spanish Ministry of Economy and Competitiveness through the Cajal Blue Brain (C080020-09; the Spanish partner of the EPFL’s Blue Brain initiative) and TIN2013-41592-P projects and by the Regional Government of Madrid through the S2013/ICE-2845-CASI-CAM-CM project. The authors thankfully acknowledge the computer resources, technical expertise and assistance provided by the Supercomputing and Visualization Center of Madrid (CeSViMa). L.A.-S. acknowledges support from the Spanish MINECO scholarship at the Residencia de Estudiantes.

Author information

Authors and Affiliations

  1. Departamento de Inteligencia Artificial, Escuela Técnica Superior de Ingenieros Informáticos, Universidad Politécnica de Madrid, Madrid, Spain

    Laura Anton-Sanchez, Concha Bielza & Pedro Larrañaga

Authors
  1. Laura Anton-Sanchez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Concha Bielza
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pedro Larrañaga
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Laura Anton-Sanchez.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Anton-Sanchez, L., Bielza, C. & Larrañaga, P. Network design through forests with degree- and role-constrained minimum spanning trees. J Heuristics 23, 31–51 (2017). https://doi.org/10.1007/s10732-017-9323-3

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10732-017-9323-3

Keywords

Search

Navigation