On metro-line crossing minimization

We consider the problem of drawing a set of simple paths along the edges of an embedded underlying graph G = (V;E) so that the total number of crossings among pairs of paths is minimized. This problem arises when drawing metro maps, where the embedding of G depicts the structure of the underlying network, the nodes of G correspond to train stations, an edge connecting two nodes implies that there exists a railway track connecting them, whereas the paths illustrate the metro lines connecting terminal stations. We call this the metro-line crossing minimization problem (MLCM). We examine several variations of the problem for which we develop algorithms that yield optimal solutions.