Breaking cycles for minimizing crossings

Abstract

We consider the one-sided crossing minimization problem (CP): given a bipartite graph G and a permutation x 0 of the vertices on a layer, find a perumuation x 1 of the vertices on the other layer which minimizes the number of edge crossings in any straightline drawing of G where vertices are placed on two parallel lines and sorted according to x 0 and x 1 . Solving CP represents a fundamental step in the construction of aesthetically pleasing layouts of heirarchies and directed graphs, but unfortunately this problem has been proved to be NP-complete. In this paper we address the strong relation between CP and the problem of computing minimum feedback arc sets in directed graphs and we devise a enw approximation algorithm for CP, called PM, that exploits this dependency. We experimantally and visually compare the performance of PM with the performance of well-known algorithms and of recent attractive strategies. Experiments are carried out on different families of randomly generated graphs, on pathological instances, and on real test sets. Performance indicators include both number of edge crossings and running time, as well as structural measures of the problem instances. We found CP to be a very interesting and rich problem from a combinatorial point of view. Our results clearly separate the behavior of the algorithms, proving the effectiveness of PM on most test sets and showing tradeoffs between quality of the solutions and running time. However, if the visual complexity of the drawings is considered, we found no clear winner. This confirms the importance of optimizing also other aesthetic criteria such as symmetry, edge length, and angular resolution.