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GRASP and Path Relinking for 2-Layer Straight Line Crossing Minimization MANUEL LAGUNA Graduate School of Business, University of Colorado, Campus Box 419, Boulder, CO 80309 [email protected] RAFAEL MARTÍ Departamento de Estadística e Investigación Operativa Facultad de Matemáticas, Universidad de Valencia Dr. Moliner 50, 46100 Burjassot (Valencia) Spain [email protected] Submitted: May 21, 1997 First revision: February 24, 1998 Second revision: June 10, 1998 Final version: August 21, 1998 ABSTRACT — In this paper, we develop a greedy randomized adaptive search procedure (GRASP) for the problem of minimizing straight-line crossings in a 2-layer graph. The procedure is fast and is particularly appealing when dealing with low-density graphs. When a modest increase in computational time is allowed, the procedure may be coupled with a path relinking strategy to search for improved outcomes. Although the principles of path relinking have appeared in the tabu search literature, this search strategy has not been fully implemented and tested. We perform extensive computational experiments with more than 3,000 graph instances to first study the effect of changes in critical search parameters and then to compare the efficiency of alternative solution procedures. Our results indicate that graph density is a major influential factor on the performance of a solution procedure.
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Laguna and Martí / 2 The problem of minimizing straight-line crossings in layered graphs has been the subject of study for at least 17 years, beginning with the Relative Degree Algorithm introduced by Carpano [2] . The problem consists of aligning the two shores V 1 and V 2 of a bipartite graph G = ( V 1 , V 2 , E ) on two parallel straight lines (layers) such that the number of crossing between the edges in E is minimized when the edges are drawn as straight lines connecting the end-nodes (Jünger and Mutzel, 1997). The problem is also known as the bipartite drawing problem (or BDP). In the BDP the problem consists of finding an optimal ordering for the vertices in both layers, which differ from the layer permutation problem (LPP) that seeks the optimal ordering of one layer only. Table 1 summarizes some of the relevant work in the area to the present. The research listed in Table 1 combines procedures specifically designed for both 2-layer and multi-layer graphs. In some instances, however, LPP procedures have been extended to the BDP case, in a similar way that 2-layer graph methods have been adapted to the multi-layer case. Table 1 Summary of relevant literature. Reference Procedure Comments Carpano [2] Relative degree algorithm Sugiyama, et al. [21] Barycenter Similar to Carpano’s. Eades and Kelly [4] Greedy insertion Splitting Averaging Greedy switching Same as barycenter. Eades and Wormald [6] Median Sugiyama [20] Rowe, et al.
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