Journal of Heuristics

, Volume 17, Issue 1, pp 39–60 | Cite as

Memetic algorithm for the antibandwidth maximization problem



The antibandwidth maximization problem (AMP) consists of labeling the vertices of a n-vertex graph G with distinct integers from 1 to n such that the minimum difference of labels of adjacent vertices is maximized. This problem can be formulated as a dual problem to the well known bandwidth problem. Exact results have been proved for some standard graphs like paths, cycles, 2 and 3-dimensional meshes, tori, some special trees etc., however, no algorithm has been proposed for the general graphs. In this paper, we propose a memetic algorithm for the antibandwidth maximization problem, wherein we explore various breadth first search generated level structures of a graph—an imperative feature of our algorithm. We design a new heuristic which exploits these level structures to label the vertices of the graph. The algorithm is able to achieve the exact antibandwidth for the standard graphs as mentioned. Moreover, we conjecture the antibandwidth of some 3-dimensional meshes and complement of power graphs, supported by our experimental results.


Memetic algorithms Antibandwidth Breadth first search 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Burkard, R.E., Donnani, H., Lin, Y., Rote, G.: The obnoxious center problem on a tree. SIAM J. Discrete Math. 14(4), 498–509 (2001) CrossRefMathSciNetMATHGoogle Scholar
  2. Calamoneri, T., Massini, A., Torok, L., Vrt’o, I.: Antibandwidth of complete k-ary trees. Electron. Notes Discrete Math. 24, 259–266 (2006) CrossRefMathSciNetGoogle Scholar
  3. Cappanera, P.: A survey on obnoxious facility location problems. Technical Report TR-99-11, Dipartimento di Informatica, Uni. di Pisa (1999) Google Scholar
  4. Dawkins, R.: The Selfish Gene. Clarendon Press, Oxford (1976) Google Scholar
  5. Diaz, J., Petit, J., Serna, M.: A survey of graph layout problems. ACM Comput. Surv. 34, 313–356 (2002) CrossRefGoogle Scholar
  6. Dobrev, S., Pardubska, D., Kralovic, Torok, L., Vrt’o, I.: Antibandwidth and cyclic antibandwidth of hamming graphs. Electron. Notes Discrete Math. 34, 295–300 (2009) CrossRefMathSciNetGoogle Scholar
  7. Donnely, S., Isaak, G.: Hamiltonian powers in threshold and arborescent comparability graphs. Discrete Math. 202, 33–44 (1999) CrossRefMathSciNetGoogle Scholar
  8. Gobel, F.: The separation number. Ars Comb. 37, 262–274 (1994) MathSciNetGoogle Scholar
  9. Hale, W.K.: Frequency assignment: theory and applications. In: Proceedings of IEEE, vol. 60, pp. 1497–1514 (1980) Google Scholar
  10. Isaak, G.: Powers of Hamiltonian paths in interval graphs. J. Graph Theory 28, 31–38 (1998) CrossRefMathSciNetMATHGoogle Scholar
  11. Leung, J.Y.-T., Vornberger, O., Witthoff, J.D.: On some variants of the bandwidth minimization problem. SIAM J. Comput. 13, 650–667 (1984) CrossRefMathSciNetMATHGoogle Scholar
  12. Miller, Z., Pritikin, D.: On the separation number of a graph. Networks 19, 651–666 (1989) CrossRefMathSciNetMATHGoogle Scholar
  13. Moscato, P.: On evolution, search, optimization, genetic algorithms and martial arts: towards memetic algorithms. In: Caltech Concurrent Computation Program, C3P Report 826 (1989) Google Scholar
  14. Ping, Y.: The separation numbers of two classes of graphs. J. Changsha Univ. Lectr. Power Nat. Sci. Ed. 12, 421–426 (1997) Google Scholar
  15. Raspaud, A., Schroder, H., Sýkora, O., Torok, L., Vrt’o, I.: Antibandwidth and cyclic antibandwidth of meshes and hypercubes. Discrete Math. 309, 3541–3552 (2009) CrossRefMathSciNetMATHGoogle Scholar
  16. Roberts, F.S.: New directions in graph theory. Ann. Discrete Math. 55, 13–44 (1993) CrossRefGoogle Scholar
  17. Torok, L., Vrt’o, I.: Antibandwidth of 3-dimensional meshes. Electron. Notes Discrete Math. 28, 161–167 (2007) CrossRefMathSciNetGoogle Scholar
  18. Wang, X., Wu, X., Dumitrescu, S.: On explicit formulas for bandwidth and antibandwidth of hypercubes. Discrete Appl. Math. 157(8), 1947–1952 (2009) CrossRefMathSciNetMATHGoogle Scholar
  19. Weili, Y., Ju, Z., Xiaoxu, L.: Dual bandwidth of some special trees. J. Zhengzhou Univ. Nat. Sci. Ed. 35 (2003) Google Scholar
  20. Yixun, L., JinJiang, Y.: The dual bandwidth problem for graphs. J. Zhengzhou Univ. Nat. Sci. Ed. 35, 1–5 (2003) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsDayalbagh Educational InstituteAgraIndia

Personalised recommendations