GRASP and Path Relinking for 2-Layer Straight Line Crossing Minimization

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. 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. The problem consists of aligning the two shores V1 and V2 of a bipartite graph G = (V1, V2, 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 Relative degree algorithm Sugiyama, et al. Barycenter Similar to Carpano’s. Eades and Kelly Greedy insertion Splitting Averaging Greedy switching Same as barycenter. Eades and Wormald Median Sugiyama Rowe, et al. Barycenter / median Variations and extensions to the barycenter / median approach. Makinen Average median Semi median Barycenter / median hybrids. Gansner, et al. Eades, et al. Barycenter / median Variations and extensions to the barycenter / median approach to multi-layered graphs. Catarci Assignment Jünger and Mutzel Branch and cut Includes a comparative study of heuristic approaches. Valls, et al. Branch and bound Small instances. Valls, et al. Martí Tabu thresholding Tabu search High quality results with long computational times. Martí and Laguna Several existing methods Empirical analysis of 16 heuristics. Laguna, et al. Tabu search Multi-layer graphs. The main application of these problems is found in automated drawing systems, where drawing speed is a critical factor. This is why for many years researchers were interested only in designing simple heuristic procedures and sacrificed solution quality for speed. The motivation for our current development is to provide high quality solutions to the 2-layer straight line cross minimization problem within a computational time that approaches simple heuristics. We will show that simple heuristics are very fast but result in inferior solutions, while high-quality solutions have been found with meta-heuristics (such as tabu search) that demand an impractical amount of computer time. Hence, our goal is to develop a procedure that can compete in speed with the simple heuristics and in quality with the complex meta-heuristics. In some of our experiments, we target low-density graphs because we believe these graph types can be found in practical applications but typically have not been Laguna and Martí / 3 used for testing proposed procedures. Also, as shown in the comparative study by Jünger and Mutzel, the performance of existing heuristics seems to quickly deteriorate as the graphs become sparser. In particular, we adapt a greedy randomized search procedure (GRASP) in this context. In addition, we explore the adaptation of a search strategy called path relinking within the GRASP framework. The path relinking strategy has been suggested in connection with tabu search, however, to the best of our knowledge, it has not been implemented and thoroughly tested. Before describing our basic procedure and its variants, we provide background on both GRASP and path relinking in the following two sections.