Experiments on exact crossing minimization using column generation

Abstract

The crossing number of a graph G is the smallest number of edge crossings in any drawing of G into the plane. Recently, the first branch-and-cut approach for solving the crossing number problem has been presented in [3]. Its major drawback was the huge number of variables out of which only very few were actually used in the optimal solution. This restricted the algorithm to rather small graphs with low crossing number. In this paper we discuss two column generation schemes; the first is based on traditional algebraic pricing, and the second uses combinatorial arguments to decide whether and which variables need to be added. The main focus of this paper is the experimental comparison between the original approach, and these two schemes. We also compare these new results to the solutions of the best known crossing number heuristic. © Springer-Verlag Berlin Heidelberg 2006.