Exact crossing minimization

Authors:
Christoph Buchheim;Dietmar Ebner;Michael Jünger;Gunnar W. Klau;Petra Mutzel;René Weiskircher
Affiliations:
Department of Computer Science, University of Cologne;Institute of Computer Graphics and Algorithms, Vienna University of Technology;Department of Computer Science, University of Cologne;Department of Mathematics and Computer Science, FU Berlin;Department of Computer Science, University of Dortmund;CSIRO Mathematical and Information Sciences
Venue:
GD'05 Proceedings of the 13th international conference on Graph Drawing
Year:
2005

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Abstract

The crossing number of a graph is the minimum number of edge crossings in any drawing of the graph into the plane. This very basic property has been studied extensively in the literature from a theoretic point of view and many bounds exist for a variety of graph classes. In this paper, we present the first algorithm able to compute the crossing number of general sparse graphs of moderate size and present computational results on a popular benchmark set of graphs. The approach uses a new integer linear programming formulation of the problem combined with strong heuristics and problem reduction techniques. This enables us to compute the crossing number for 91 percent of all graphs on up to 40 nodes in the benchmark set within a time limit of five minutes per graph.