Break the \ Right " Cycles and Get the \ Best

We consider the two-layer crossing minimization problem (CP): given a two-layered graph and a permutation of the vertices on a layer, nd a permutation of the vertices on the other layer so that the number of edge crossings is minimum. Solving CP represents a fundamental step in the construction of aesthetically pleasing layouts of hierarchies and directed graphs. As CP has been proved to be NP-complete, several heuristics for it have been designed, but unfortunately, according to previous experimental results, even algorithms with good theoretical performances appear not to be good in practice. The contribution of this paper is twofold. First, we present new theoretical results that allow us to design a polynomial-time approximation algorithm for CP (in short, PM) based on the computation of feedback arc sets. Furthermore, we perform an experimental analysis of algorithm PM both on randomly generated bipartite graphs and on pathological instances for CP. In particular, we compare its performances, with respect to the number of edge crossings and to the running time, to the performances of three classical algorithms for CP: the barycenter heuristic, an enhanced version of the median algorithm, and a recent attractive greedy strategy. Our experimental results show that algorithm PM stands out as a practical and eeective method for minimizing crossings in bipartite drawings.