Variable neighborhood search with ejection chains for the antibandwidth problem

Abstract

In this paper, we address the optimization problem arising in some practical applications in which we want to maximize the minimum difference between the labels of adjacent elements. For example, in the context of location models, the elements can represent sensitive facilities or chemicals and their labels locations, and the objective is to locate (label) them in a way that avoids placing some of them too close together (since it can be risky). This optimization problem is referred to as the antibandwidth maximization problem (AMP) and, modeled in terms of graphs, consists of labeling the vertices with different integers or labels such that the minimum difference between the labels of adjacent vertices is maximized. This optimization problem is the dual of the well-known bandwidth problem and it is also known as the separation problem or directly as the dual bandwidth problem. In this paper, we first review the previous methods for the AMP and then propose a heuristic algorithm based on the variable neighborhood search methodology to obtain high quality solutions. One of our neighborhoods implements ejection chains which have been successfully applied in the context of tabu search. Our extensive experimentation with 236 previously reported instances shows that the proposed procedure outperforms existing methods in terms of solution quality.

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References

  1. Aiex, R.M., Resende, M.G.C., Ribeiro, C.C.: Probability distribution of solution time in GRASP: An experimental investigation. J. Heuristics 8, 343–373 (2002)

    MATH  Article  Google Scholar 

  2. Bansal, R., Srivastava, K.: Memetic algorithm for the antibandwidth maximization problem. J. Heuristics 17, 39–60 (2011)

    MATH  Article  Google Scholar 

  3. Burkard, R.E., Donnani, H., Lin, Y., Rote, G.: The obnoxious center problem on a tree. SIAM J. Discret. Math. 14(4), 498–590 (2001)

    MATH  Article  Google Scholar 

  4. Cappanera, P.: A survey on obnoxious facility location problems. Technical Report TR-99-11, Dipartimento di Informatica, Università di Pisa (1999)

  5. Díaz, J., Petit, J., Serna, M.: A survey of graph layout problems. J. ACM Comput. Surv. 34(3), 313–356 (2002)

    Article  Google Scholar 

  6. Dobrev, S., Královic, R., Pardubská, D., Török, L., Vrt’o, I.: Antibandwidth and cyclic antibandwidth of Hamming graphs. Electron. Notes Discret. Math. 34, 295–300 (2009)

    Article  Google Scholar 

  7. Duarte, A., Martí, R., Resende, M.G.C., Silva, R.M.A.: GRASP with path relinking heuristics for the antibandwidth problem. Networks 58(3), 171–189 (2011)

    MathSciNet  MATH  Article  Google Scholar 

  8. Duarte, A., Escudero, L.F., Martí, R., Mladenovic, N., Pantrigo, J.J., Sánchez-Oro, J.: Variable neighborhood search for the vertex separation problem. Comput. Oper. Res. 39(12), 3247–3255 (2012)

    MathSciNet  Article  Google Scholar 

  9. Glover, F., Laguna, M.: Tabu Search. Kluwer, Norwell (1997)

    Google Scholar 

  10. Hale, W.K.: Frequency assignment: theory and applications. Proc. IEEE 68, 1497–1514 (1980)

    Article  Google Scholar 

  11. Hansen, P., Mladenovic, N., Brimberg, J., Moreno-Pérez, J.A.: Variable neighborhood search. In: Gendreau, M., Potvin, J.-Y. (eds.) Handbook of Metaheuristics, 2nd edn, pp. 61–86. Springer, Heidelberg (2010)

  12. Harwell-Boeing: http://math.nist.gov/MatrixMarket/data/Harwell-Boeing/ (2011)

  13. Leung, J.Y.-T., Vornberger, O., Witthoff, J.D.: On some variants of the bandwidth minimization problem. SIAM J. Comput. 13, 650–667 (1984)

    MathSciNet  MATH  Article  Google Scholar 

  14. Martí, R., Duarte, A., Laguna, M.: Advanced scatter search for the max-cut problem. INFORMS J. Comput. 21(1), 26–38 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  15. Miller, Z., Pritikin, D.: On the separation number of a graph. Networks 19, 651–666 (1989)

    MathSciNet  MATH  Article  Google Scholar 

  16. Pantrigo, J.J., Martí, R., Duarte, A., Pardo, E.G.: Scatter search for the cutwidth minimization problem. Ann. Oper. Res. 199(1), 285–304 (2012)

    Google Scholar 

  17. Piñana, E., Plana, I., Campos, V., Martí, R.: GRASP and path relinking for the matrix bandwidth minimization. Eur. J. Oper. Res. 153, 200–210 (2004)

    MATH  Article  Google Scholar 

  18. Raspaud, A., Schröder, H., Sykora, O., Török, L., Vrt’o, I.: Antibandwidth and cyclic antibandwidth of meshes and hypercubes. Discret. Math. 309, 3541–3552 (2009)

    MATH  Article  Google Scholar 

  19. Rego, C.: Node ejection chains for the vehicle routing problem: sequential and parallel algorithms. Parallel Comput. 27(3), 201–222 (2001)

    MATH  Article  Google Scholar 

  20. Resende, M., Martí, R., Gallego, M., Duarte, A.: GRASP and path relinking for the max-min diversity problem. Comput. Oper. Res. 37, 498–508 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  21. Rodriguez-Tello, E., Jin-Kao, H., Torres-Jimenez, J.: An effective two-stage simulated annealing algorithm for the minimum linear arrangement problem. Comput. Oper. Res. 35(10), 3331–3346 (2008)

    MATH  Article  Google Scholar 

  22. Török, L., Vrt’o, I.: Antibandwidth of 3-dimensional meshes. Electron. Notes Discret. Math. 28, 161–167 (2007)

    Article  Google Scholar 

  23. Yixun, L., Jinjiang, Y.: The dual bandwidth problem for graphs. J. Zhengzhou Univ. 35, 1–5 (2003)

    MATH  Google Scholar 

  24. Wang, X., Wu, X., Dumitrescu, S.: On explicit formulas for bandwidth and antibandwidth of hypercubes. Discret. Appl. Math. 157(8), 1947–1952 (2009)

    MathSciNet  MATH  Article  Google Scholar 

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Acknowledgements

This research has been partially supported by the Ministerio de Ciencia e Innovación of Spain within the OPTSICOM project (http://www.optsicom.es/) with grant codes TIN2008-05854, TIN2009-07516, TIN2012-35632, and P08-TIC-4173.

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Correspondence to Rafael Martí.

Appendix

Appendix

Tables 67 and 8 report the comparison of the state-of-the-art methods over the set of instances with unknown optimum. We consider a time limit of 150 s per instance. Each table shows for each instance the best known value, Best val., the tightest upper bound (Yixun and Jinjiang 2003), UB, the relative deviation (in percentage) between the best known value and the upper bound, Dev, and finally the heuristic method (or methods) achieving these results.

Table 6 Best values, best methods and upper bounds per instance for Harwell-Boeing matrices
Table 7 Best values, best methods and upper bounds per instance for 3D grids
Table 8 Best values, best methods and upper bounds per instance for Caterpillars

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Lozano, M., Duarte, A., Gortázar, F. et al. Variable neighborhood search with ejection chains for the antibandwidth problem. J Heuristics 18, 919–938 (2012). https://doi.org/10.1007/s10732-012-9213-7

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Keywords

  • Metaheuristics
  • VNS
  • Layout problems