Two-Layer Planarization in Graph Drawing

We study the \tlpp s 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 \cite{Mut97} using polyhedral combinatorics. Here, we study the remaining two cases and the relationships between the associated polytopes. In particular, we investigate the polytope $\calp _1$ associated with the two-layer {\em planarization} problem with one fixed layer. We provide an overview on the relationships between $\calp _1$ and the polytope $\calq _1$ associated with the two-layer {\em crossing minimization} problem with one fixed layer, the linear ordering polytope, the \tlpp\ with zero and two layers fixed. We will see that all facet-defining inequalities in $\calq _1$ are also facet-defining for $\calp _1$. 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 \tlpp\ 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 $\calp _2$, which is useful in our branch-and-cut algorithm for the one-layer fixed case