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Tabu search tutorial. A Graph Drawing Application

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Abstract

Tabu search is an optimization methodology that guides a local heuristic search procedure to explore the solution space beyond local optimality. It is substantiated by the hypothesis that an intelligent solving algorithm must incorporate memory to base its decisions on information collected during the search. The method creates in this way a learning pattern to explore the solution space economically and effectively. Tabu search is a metaheuristic that has proved its effectiveness in a wide variety of problems, especially in combinatorial optimization. We provide here a practical description of the methodology and apply it to a novel graph drawing problem. The most popular method of drawing graphs is the Sugiyama’s framework, which obtains a drawing of a general graph by transforming it into a proper hierarchy. In this way, the number of edge crossing is minimized in the first stage of the procedure. Many metaheuristics have been proposed to solve the crossing minimization problem within this drawing convention. The second stage of this procedure minimizes the number of bends of long arcs without increasing the number of crossings, thus obtaining a readable drawing. In this paper, we propose an alternative approach to simultaneously minimize the two criteria: crossing and long arc bends. We apply tabu search to solve this problem and compare its solutions with the optimal values obtained with CPLEX in small and medium-size instances.

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... Since these early proposals, many researchers have developed models and methods, mainly heuristics and metaheuristics (Glover et al. 2021), to provide high-quality solutions to these two problems. From Erkut and Neuman (1989) to Martínez-Gavara et al. (2021), we can find more than 50 papers published in top-ranked journals proposing solving methods for these problems and their variants, where the MaxSum model is the most widely applied. ...
... Extended MinDiff, and therefore we will use a competitive MinDiff solver as a reference in our comparisons. Porumbel et al. (2011) proposed a fast tabu search (Glover et al. 2021) for a model that combines the MaxMin and the MaxSum problems, which can be considered a first approach to solve the Extended MaxSum. In particular, the authors minimize the MaxMin objective function and consider the MaxSum as a secondary objective. ...
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... The tabu search algorithm (TS) is a metaheuristic and is widely used in solving optimization problems [27], [28]. Previous studies [29]- [33] applied the TS to optimize manufacturing system resources and verified the effectiveness of the TS in allocating buffers into manufacturing systems. ...
... Tabu tenure is the length of the tabu list, which has an important effect on the solution quality of the TS [27]. To improve computational efficiency, the dynamic rule for the tabu tenure may be useful. ...
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We propose a metaheuristic algorithm for the multi-resource generalized assignment problem (MRGAP). MRGAP is a generalization of the generalized assignment problem, which is one of the representative combinatorial optimization problems known to be NP-hard. The algorithm features a very large-scale neighborhood search, which is a mechanism of conducting the search with complex and powerful moves, where the resulting neighborhood is efficiently searched via the improvement graph. We also incorporate an adaptive mechanism for adjusting search parameters, to maintain a balance between visits to feasible and infeasible regions. Computational comparisons on benchmark instances show that the method is effective, especially for types D and E instances, which are known to be quite difficult.
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We present a simple, linear-time algorithm to determine horizontal coordinates in layered layouts subject to a given ordering within each layer. The algorithm is easy to implement and compares well with existing approaches in terms of assignment quality.
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Two kinds of new methods are developed to obtain effective representations of hierarchies automatically: theoretical and heuristic methods. The methods determine the positions of vertices in two steps. First the order of the vertices in each level is determined to reduce the number of crossings of edges. Then horizontal positions of the vertices are determined to improve further the readability of drawings. The theoretical methods are useful in recognizing the nature of the problem, and the heuristic methods make it possible to enlarge the size of hierarchies with which we can deal. Performance tests of the heuristic methods and several applications are presented.
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A study of questions raised by the conception of a tool for computer-aided decision analysis that would facilitate interactive structural systems analysis and the tool itself, named GT1VX, are presented. Focusing on systems known by their elements and the relations among them, two hierarchies that apply to graphs of different types: first, rank hierarchy that is adapted to digraphs with cycles and second, number hierarchy that applies to strongly connected digraphs and to nondirected graphs are presented. Standard forms of representation of these hierarchies are defined, and the question of reducing the number of edge crossings in the drawings is studied. Finally, the software GT1VX is described. It was developed for the automatic display of graphs hierarchized according to the rank hierarchy and to the number hierarchy; GT1VX yields, within agreeable computing times, drawings in the standard forms of representation that have a limited number of edge crossings. Three real-life examples of the application of this tool are also presented.
Parametric combinations of local job shop rules
  • F Glover
Glover F (1963) Parametric combinations of local job shop rules. Chapter IV, ONR Research Memorandum No. 117, GSIA, Carnegie-Mellon University, Pittsburgh
Tabu search and adaptive memory programming-advances, applications and challenges
  • F Glover
Glover F (1997) Tabu search and adaptive memory programming-advances, applications and challenges. In: Barr RS, Helgason RV, Kennington JL (eds) Interfaces in computer science and operations research. Kluwer Academic Publishers, Boston, pp 1-75
A simple multi-wave algorithm for the uncapacitated facility location problem
  • F Glover
  • S Hanafi
  • O Guermi
  • I Crevits
Glover F, Hanafi S, Guermi O, Crevits I (2018a) A simple multi-wave algorithm for the uncapacitated facility location problem. Front Eng Manag 5:451-465