A New Approach to Exact Crossing Minimization

Authors:
Markus Chimani;Petra Mutzel;Immanuel Bomze
Affiliations:
Faculty of Computer Science, Dortmund University of Technology, Germany;Faculty of Computer Science, Dortmund University of Technology, Germany;Dep. of Statistics and Decision Support Systems, University of Vienna, Austria
Venue:
ESA '08 Proceedings of the 16th annual European symposium on Algorithms
Year:
2008

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Abstract

The crossing numberproblem is to find the smallest number of edge crossings necessary when drawing a graph into the plane. Eventhough the problem is NP-hard, we are interested in practically efficient algorithms to solve the problem to provable optimality. In this paper, we present a novel integer linear programming (ILP) formulation for the crossing number problem. The former formulation [4] had to transform the crossing number polytope into a higher-dimensional polytope. The key idea of our approach is to directly consider the natural crossing number polytope and cut it with multiple linear-ordering polytopes. This leads to a more compact formulation, both in terms of variables and constraints.We describe a Branch-and-Cut algorithm, together with a combinatorial column generation scheme, in order to solve the crossing number problem to provable optimality. Our experiments show that the new approach is more effective than the old one, even when considering a heavily improved version of the former formulation (also presented in this paper). For the first time, we are able to solve graphs with a crossing number of up to 37.