Ordering Metro Lines by Block Crossings

A problem that arises in drawings of transportation networks is to minimize the number of crossings between different transportation lines. While this can be done efficiently under specific constraints, not all solutions are visually equivalent. We suggest merging single crossings into block crossings, that is, crossings of two neighboring groups of consecutive lines. Unfortunately, minimizing the total number of block crossings is NP-hard even for very simple graphs. We give approximation algorithms for special classes of graphs and an asymptotically worst-case optimal algorithm for block crossings on general graphs. Furthermore, we show that the problem remains NP-hard on planar graphs even if both the maximum degree and the number of lines per edge are bounded by constants; on trees, this restricted version becomes tractable. Submitted: March 2014 Reviewed: August 2014 Revised: January 2015 Accepted: January 2015 Final: February 2015 Published: February 2015 Article type: Regular Paper Communicated by: H. Meijer Some of the results in this paper have been presented at the 38th International Symposium on Mathematical Foundations of Computer Science (MFCS 2013) [16]. E-mail addresses: [email protected] (Martin Fink) [email protected] (Sergey Pupyrev) 112 Fink et al. Ordering Metro Lines by Block Crossings (a) 12 pairwise crossings. (b) 12 pairwise crossings grouped into 3 block crossings. Figure 1: Optimal orderings of a metro network.