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Abstract:
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.
