An ILP for the metro-line crossing problem

Authors:
Matthew Asquith;Joachim Gudmundsson;Damian Merrick
Affiliations:
Sophos Pty Ltd, North Sydney, Australia and University of Sydney, Australia;NICTA, Sydney, Australia;University of Sydney, Australia and NICTA, Sydney, Australia
Venue:
CATS '08 Proceedings of the fourteenth symposium on Computing: the Australasian theory - Volume 77
Year:
2008

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Abstract

In this paper we consider a problem that occurs when drawing public transportation networks. Given an embedded graph G = (V, E) (e.g. the railroad network) and a set H of paths in G (e.g. the train lines), we want to draw the paths along the edges of G such that they cross each other as few times as possible. For aesthetic reasons we insist that the relative order of the paths that traverse a vertex does not change within the area occupied by the vertex. We prove that the problem, which is known to be NP-hard, can be rewritten as an integer linear program that finds the optimal solution for the problem. In the case when the order of the endpoints of the paths is fixed we prove that the problem can be solved in polynomial time. This improves a recent result by Bekos et al. (2007).