A New Heuristic Algorithm for the Vertex Separation Problem

  • Norberto Castillo-GarcíaEmail author
  • Paula Hernández Hernández
Part of the Studies in Computational Intelligence book series (SCI, volume 749)


The Vertex Separation Problem (VSP) belongs to a family of graph layout problems. VSP consists in finding a linear ordering of the vertices of a graph such that the maximum number of vertex separators at each position of the ordering is minimized. This problem has important practical applications in fields such as very large scale integration design, computer language compiler design or natural language processing. VSP has been proven to be NP-hard. In the literature reviewed, we found several heuristic and metaheuristic algorithms designed for solving large instances of VSP. As far as we are aware, these algorithms do not use fuzzy logic. In this chapter, we adapt two fuzzy logic classifiers (FLC) to a constructive algorithm from the literature. More precisely, the first FLC is used to select the vertex to be placed at the first position of the linear ordering according to the adjacency degree. The second FLC is used to select the following vertices according to the number of vertex separators. We have designed five variants of our fuzzy heuristic. The computational experiment indicates that the first four variants have a similar behavior in solution quality and execution time.


Vertex separation problem Heuristics Constructive algorithms Fuzzy logic classifier 



The first author thanks Tecnológico Nacional de México and especially Instituto Tecnológico de Altamira for their support in this research. The second author would like to thank the CATEDRAS CONACYT program.


  1. 1.
    J. Díaz, J. Petit, M. Serna, A survey of graph layout problems. ACM Comput. Surv. (CSUR) 34(3), 313–356 (2002)CrossRefGoogle Scholar
  2. 2.
    T. Lengauer, Black-white pebbles and graph separation. Acta Informatica 16(4), 465–475 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    C.E. Leiserson, Area-efficient graph layouts, in 21st Annual Symposium on Foundations of Computer Science, 1980, (IEEE, 1980, October), pp. 270–281Google Scholar
  4. 4.
    H. Bodlaender, J. Gustedt, J.A. Telle, Linear-time register allocation for a fixed number of registers. In SODA, vol. 98 (1998, January), pp. 574–583Google Scholar
  5. 5.
    A. Kornai, Z. Tuza, Narrowness, pathwidth, and their application in natural language processing. Discrete Appl. Math. 36(1), 87–92 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    I.C. Lopes, J.M. Carvalho, Minimization of open orders using interval graphs. Int. J. Appl. Math. 40(4), 297–306 (2010)MathSciNetzbMATHGoogle Scholar
  7. 7.
    G. Luque, E. Alba, Metaheuristics for the DNA fragment assembly problem. Int. J. Comput. Itell. Res. 1(2), 98–108 (2005)Google Scholar
  8. 8.
    H.J.F. Huacuja, N. Castillo-García, Optimization of the Vertex Separation Problem with genetic algorithms. In Handbook of Research on Military, Aeronautical, and Maritime Logistics and Operations (IGI Global, 2016), pp. 13–31Google Scholar
  9. 9.
    A. Duarte, L.F. Escudero, R. Martí, N. Mladenovic, J.J. Pantrigo, J. Sánchez-Oro, Variable neighborhood search for the Vertex Separation Problem. Comput. Oper. Res. 39(12), 3247–3255 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    J. Sánchez-Oro, J.J. Pantrigo, A. Duarte, Combining intensification and diversification strategies in VNS. An application to the Vertex Separation Problem. Comput. Oper. Res. 52, 209–219 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    N. Castillo-García, H.J.F. Huacuja, R.A.P. Rangel, J.A.M. Flores, J.J.G. Barbosa, J.M.C. Valadez, On the exact solution of VSP for general and structured graphs: models and algorithms, in Recent Advances on Hybrid Approaches for Designing Intelligent Systems (Springer International Publishing, 2014), pp. 519–532Google Scholar
  12. 12.
    N. Castillo-García, H.J.F. Huacuja, R.A.P. Rangel, J.A.M. Flores, J.J.G. Barbosa, J.M.C. Valadez, Comparative study on constructive heuristics for the Vertex Separation Problem, in Design of Intelligent Systems Based on Fuzzy Logic, Neural Networks and Nature-Inspired Optimization (Springer International Publishing, 2015), pp. 465–474Google Scholar
  13. 13.
    H. Fraire Huacuja, N. Castillo-García, R.A. Pazos Rangel, J.A. Martínez Flores, J.J. González Barbosa, J.M. Carpio Valadez, Two new exact methods for the Vertex Separation Problem. Int. J. Comb. Optim. Prob. Inform. 6(1), 31–41 (2015)Google Scholar
  14. 14.
    H.J. Fraire-Huacuja, N. Castillo-García, M.C. López-Locés, J.A.M. Flores, J.J.G. Barbosa, J.M.C. Valadez, Integer linear programming formulation and exact algorithm for computing pathwidth, in Nature-Inspired Design of Hybrid Intelligent Systems (Springer International Publishing, 2017), pp. 673–686Google Scholar
  15. 15.
    R. Sepúlveda, O. Montiel, O. Castillo, P. Melin, Fundamentos de Lógica Difusa. Ediciones ILCSA (2002)Google Scholar
  16. 16.
    J.J. Pantrigo, R. Martí, A. Duarte, E.G. Pardo, Scatter search for the cutwidth minimization problem. Ann. Oper. Res. 199(1), 285–304 (2012)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  • Norberto Castillo-García
    • 1
    Email author
  • Paula Hernández Hernández
    • 2
  1. 1.Tecnológico Nacional de México, Instituto Tecnológico de AltamiraAltamiraMexico
  2. 2.CONACYT-Universidad Autónoma de Tamaulipas-Facultad de IngenieríaTampicoMexico

Personalised recommendations