# The Constrained Crossing Minimization Problem

## Abstract

In this paper we consider the *constrained crossing minimization problem defined* as follows. Given a connected planar graph *G = (V,E)*, a combinatorial embedding *II(G)* of *G*, and a set of pairwise distinct edges *F ⊆ V × V*, find a drawing of *G* ^{′} = (*V,E ∼ F*) such that the combinatorial embedding *II(G)* of *G* is preserved and the number of edge crossings is minimized. The constrained crossing minimization problem arises in the graph drawing method based on planarization. In [4] we have shown that we can formulate the constrained crossing minimization problem as an *|F|*-pairs shortest walks problem, where we want to minimize the sum of the lengths of the walks plus the number of crossings between the walks

Here we present an integer linear programming formulation (ILP) for the *shortest crossing walks problem*. Furthermore, we will present additional valid inequalities that strengthen the formulation. Based on our results we have designed and implemented a branch and cut algorithm. Our computational experiments for the constrained crossing minimization problem on a benchmark set of graphs ([1]) are encouraging. This is the first time that practical instances of the constrained crossing minimization problem can be solved to provable optimality.

## References

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