Fixed linear crossing minimization by reduction to the maximum cut problem

Authors:
Christoph Buchheim;Lanbo Zheng
Affiliations:
Computer Science Department, University of Cologne, Germany;School of Information Technologies, University of Sydney, Australia
Venue:
COCOON'06 Proceedings of the 12th annual international conference on Computing and Combinatorics
Year:
2006

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Abstract

Many real-life scheduling, routing and location problems can be formulated as combinatorial optimization problems whose goal is to find a linear layout of an input graph in such a way that the number of edge crossings is minimized. In this paper, we study a restricted version of the linear layout problem where the order of vertices on the line is fixed, the so-called fixed linear crossing number problem (FLCNP). We show that this $\mathcal{NP}$-hard problem can be reduced to the well-known maximum cut problem. The latter problem was intensively studied in the literature; efficient exact algorithms based on the branch-and-cut technique have been developed. By an experimental evaluation on a variety of graphs, we show that using this reduction for solving FLCNP compares favorably to earlier branch-and-bound algorithms.