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\documentclass[letterpaper]{article} \begin{document} \title{The Constrained Crossing Minimization Problem} \author{Petra Mutzel \and Thomas Ziegler\thanks{Corresponding author, email: {\tt tziegler@mpi-sb.mpg.de.} Research supported by ZFE, Siemens AG, M\"unchen.}} \date{Max-Planck-Institut f\"ur Informatik,\\ Im Stadtwald, D-66123 Saarbr\"ucken} \maketitle In this paper we investigate the {\em constrained crossing minimization problem} defined as follows. Given a connected, planar graph $G=(V,E)$, a combinatorial embedding $\Pi(G)$ of $G$, and a set of pairwise distinct edges $F\subseteq V\times V$, find a drawing of $G^\prime=(V,E\cup F)$ such that the combinatorial embedding $\Pi(G)$ of $G$ is preserved and the number of crossings is minimized. The constrained crossing minimization problem arises in the drawing method based on planarization. The constrained crossing minimization problem is NP--hard. We can formulate it as an $|F|$--pairs shortest walks problem on an extended dual graph, in which we want to minimize the sum of the lengths of the walks plus the number of crossings between walks. Here we present an integer linear programming formulation (ILP) for the {\em shortest crossing walks problem}. Furthermore we 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 are encouraging. This is the first time that practical instances of the problem can be solved to provable optimality. \end{document}