Two-Layer Planarization in Graph Drawing

We study the two-layer planarization problems that have applications in Automatic Graph Drawing. We are searching for a two-layer planar subgraph of maximum weight in a given two-layer graph. Depending on the number of layers in which the vertices can be permuted freely, that is, zero, one or two, different versions of the problems arise. The latter problem was already investigated in 11 using polyhedral combinatorics. Here, we study the remaining two cases and the relationships between the associated polytopes. In particular, we investigate the polytope P1 associated with the two-layer planarization problem with one fixed layer. We provide an overview on the relationships between P1 and the polytope Q1 associated with the two-layer crossing minimization problem with one fixed layer, the linear ordering polytope, the two-layer planarization problem with zero and two layers fixed. We will see that all facet-defining inequalities in Q1 are also facet-defining for P1. Furthermore, we give some new classes of facet-defining inequalities and show how the separation problems can be solved. First computational results are presented using a branch-and-cut algorithm. For the case when both layers are fixed, the two-layer planarization problem can be solved in polynomial time by a transformation to the heaviest increasing subsequence problem. Moreover, we give a complete description of the associated polytope P2, which is useful in our branch-and-cut algorithm for the one-layer fixed case.