Computational Optimization and Applications

, Volume 68, Issue 3, pp 775–797 | Cite as

Variable neighborhood scatter search for the incremental graph drawing problem

  • Jesús Sánchez-Oro
  • Anna Martínez-Gavara
  • Manuel Laguna
  • Rafael MartíEmail author
  • Abraham Duarte


Automated graph-drawing systems utilize procedures to place vertices and arcs in order to produce graphs with desired properties. Incremental or dynamic procedures are those that preserve key characteristics when updating an existing drawing. These methods are particularly useful in areas such as planning and logistics, where updates are frequent. We propose a procedure based on the scatter search methodology that is adapted to the incremental drawing problem in hierarchical graphs. These drawings can be used to represent any acyclic graph. Comprehensive computational experiments are used to test the efficiency and effectiveness of the proposed procedure.


Graph drawing Scatter search Incremental graph Dynamic graph drawing Metaheuristics 



This work has been partially supported by the Spanish “Ministerio de Economía y Competitividad” and by “Comunidad de Madrid,” Grants Refs. TIN2015-65460-C02 and S2013/ICE-2894, respectively. Additionally, Prof. Martinez-Gavara and Sánchez-Oro thank “Programa de Ayudas para Estancias Cortas en otras Universidades y Centros de Investigación,” Universidad de Valencia (Ref. UV-INV_EPDI16-384465) and “Ayudas a la Movilidad Predoctoral para Estancias Breves,” Ministerio de Economía y Competitividad (Ref. EEBB-I-16-11312) for supporting their visits to the University of Colorado, Boulder.


  1. 1.
    Böhringer, K.F., Paulisch, F.N.: Using constraints to achieve stability in automatic graph layout algorithms. In: Proceedings of CHI’90, pp. 43–51. ACM (1990)Google Scholar
  2. 2.
    Branke, J.: Dynamic graph drawing. In: Wagner, D., Kaufmann, M. (eds.) Drawing Graphs: Methods and Models, pp. 228–246. Springer, Berlin (2001)CrossRefGoogle Scholar
  3. 3.
    Carpano, M.: Automatic display of hierarchized graphs for computer-aided decision analysis. IEEE Trans. Syst. Man Cybern. 10(11), 705–715 (1980)CrossRefGoogle Scholar
  4. 4.
    Di Battista, G., Eades, P., Tamassia, R., Tollis, I.: Graph Drawing: Algorithms for the Visualization of Graphs, 1st edn. Prentice Hall PTR, Upper Saddle River (1998)zbMATHGoogle Scholar
  5. 5.
    Duarte, A., Pantrigo, J.J., Pardo, E.G., Sánchez-Oro, J.: Parallel variable neighbourhood search strategies for the cutwidth minimization problem. IMA J. Manag. Math. 27(1), 55–73 (2016)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Feo, T.A., Resende, M.G.: Greedy randomized adaptive search procedures. J. Glob. Optim. 6(2), 109–133 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Garey, M., Johnson, D.: Crossing number is NP-complete. SIAM J. Algebraic Discrete Methods 4(3), 312–316 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Glover, F.: Tabu search and adaptive memory programming—advances, applications and challenges. In: Barr, R.S., Helgason, R.V., Kennington, J.L. (eds.) Interfaces in Computer Science and Operations Research: Advances in Metaheuristics, Optimization, and Stochastic Modeling Technologies, pp. 1–75. Springer, Boston (1997)Google Scholar
  9. 9.
    Glover, F., Laguna, M.: Tabu Search. Kluwer Academic Publishers, Norwell (1997)CrossRefzbMATHGoogle Scholar
  10. 10.
    Hansen, P., Mladenović, N., Moreno-Pérez, J.: Variable neighborhood search: methods and applications. 4OR 6(4), 319–360 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Kaufmann, M., Wagner, D.: Drawing Graphs: Methods and Models. Lecture Notes in Computer Science. Springer, Berlin (2001)CrossRefzbMATHGoogle Scholar
  12. 12.
    Laguna, M., Martí, R.: GRASP and path relinking for 2-layer straight line crossing minimization. INFORMS J. Comput. 11, 44–52 (1999)CrossRefzbMATHGoogle Scholar
  13. 13.
    Laguna, M., Martí, R.: Scatter Search: Methodology and Implementations in C. Kluwer Academic Publisher, Norwell (2003)CrossRefzbMATHGoogle Scholar
  14. 14.
    Laguna, M., Marti, R., Valls, V.: Arc crossing minimization in hierarchical digraphs with tabu search. Comput. Oper. Res. 24(12), 1175–1186 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Martí, R., Estruch, V.: Incremental bipartite drawing problem. Comput. Oper. Res. 28(13), 1287–1298 (2001)CrossRefzbMATHGoogle Scholar
  16. 16.
    Mladenović, N., Hansen, P.: Variable neighborhood search. Comput. Oper. Res. 24(11), 1097–1100 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Mutzel, P., Jünger, M., Leipert, S.: How to layer a directed acyclic graph. In: Graph Drawing: 9th International Symposium, GD 2001 Vienna, Austria, September 23–26, 2001 Revised Papers, pp. 16–30. Springer, Berlin (2002)Google Scholar
  18. 18.
    North, S.C.: Incremental layout in DynaDAG. In: Proceedings of of Graph Drawing’95, volume 1027 of Lecture Notes in Computer Science, pp. 409–418. Springer, Berlin (1996)Google Scholar
  19. 19.
    Pinaud, B., Kuntz, P., Lehn, R.: Dynamic graph drawing with a hybridized genetic algorithm. In: Parmee, I.C. (ed.) Adaptive Computing in Design and Manufacture VI, 2004, pp. 365–375. Springer, Bristol (2004)CrossRefGoogle Scholar
  20. 20.
    Purchase, H.: Which aesthetic has the greatest effect on human understanding? In: Proceedings of Graph Drawing’97, vol 1353 de Lecture Notes in Computer Science, pp. 248–261. Springer, Berlin (1997)Google Scholar
  21. 21.
    Purchase, H.: Effective information visualization: a study of graph drawing aesthetics and algorithms. Interact. Comput. 13(2), 147–162 (2000)CrossRefGoogle Scholar
  22. 22.
    Resende, M.G., Ribeiro, C.C.: GRASP with path-relinking: recent advances and applications. In: Ibaraki, T., Nonobe, K., Yagiura, M. (eds.) Metaheuristics: Progress as Real Problem Solvers, pp. 29–63. Springer, New York (2005)CrossRefGoogle Scholar
  23. 23.
    Sugiyama, K., Tagawa, S., Toda, M.: Methods for visual understanding of hierarchical system structures. IEEE Trans. Syst. Man Cybern. (SMC) 11(2), 109–125 (1981)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Departamento de Ciencias de la ComputaciónUniversidad Rey Juan CarlosMadridSpain
  2. 2.Departamento de Estadística e Investigación OperativaUniversidad de ValenciaValenciaSpain
  3. 3.Leeds School of BusinessUniversity of Colorado BoulderBoulderUSA

Personalised recommendations