"A Branch-and-Cut Approach to the Crossing Number Problem"
Technical report by Christoph Buchheim, Markus Chimani, Dietmar Ebner, Carsten Gutwenger, Michael Jünger, Gunnar W. Klau, Petra Mutzel, René Weiskircher, available as BibTeX Source, postscript file and compressed postscript file.
Zentrum für Angewandte Informatik Köln, Lehrstuhl Jünger
Preprint Key:	zaik2006-508
Keywords:	branch-and-cut, column generation, crossing number
MSC codes:	05C10, 90C35, 90C57
This technical report has 26 pages, was written in January 2006, it has not been published.
Abstract:	The crossing number of a graph is the minimum number of edge crossings in any drawing of the graph in the plane. Extensive research has produced bounds on the crossing number and exact formulae for special graph classes, yet the crossing numbers of graphs such as K_{11} or K_{9,11} are still unknown. Finding the crossing number is NP-hard for general graphs and no practical algorithm for its computation has been published so far. We present an integer linear programming formulation that is based on a reduction of the general problem to a restricted version of the crossing number problem in which each edge may be crossed at most once. We also present cutting plane generation heuristics and a column generation scheme. As we demonstrate in a computational study, a branch-and-cut algorithm based on these techniques as well as recently published preprocessing algorithms can be used to successfully compute the crossing number for small to medium sized general graphs.