An Alternative Method to Crossing Minimization on Hierarchical Graphs

A common method for drawing directed graphs is, as a first step, to partition the vertices into a set of k levels and then, as a second step, to permute the vertices within the levels such that the number of crossings is minimized. We suggest an alternative method for the second step, namely, removing the minimal number of edges such that the resulting graph is k-level planar. For the final diagram the removed edges are reinserted into a k-level planar drawing. Hence, instead of considering the k-level crossing minimization problem, we suggest solving the k-level planarization problem. In this paper we address the case k=2. First, we give a motivation for our approach. Then, we address the problem of extracting a 2-level planar subgraph of maximum weight in a given 2-level graph. This problem is NP-hard. Based on a characterization of 2-level planar graphs, we give an integer linear programming formulation for the 2-level planarization problem. Moreover, we define and investigate the polytope $2{\cal LPS}(G)$ associated with the set of all 2-level planar subgraphs of a given 2-level graph G. We will see that this polytope has full dimension and that the inequalities occurring in the integer linear description are facet-defining for $2{\cal LPS}(G)$. The inequalities in the integer linear programming formulation can be separated in polynomial time; hence they can be used efficiently in a branch-and-cut method for solving practical instances of the 2-level planarization problem. Furthermore, we derive new inequalities that substantially improve the quality of the obtained solution. The separation problem for all the new classes of inequalities can be solved in polynomial time. We report on extensive computational results.