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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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.
Tables 6, 7 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.
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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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