GD 2008: Graph Drawing pp 336-347

# Two Polynomial Time Algorithms for the Metro-line Crossing Minimization Problem

• Evmorfia Argyriou
• Michael A. Bekos
• Michael Kaufmann
• Antonios Symvonis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5417)

## Abstract

The metro-line crossing minimization (MLCM) problem was recently introduced as a response to the problem of drawing metro maps or public transportation networks, in general. According to this problem, we are given a planar, embedded graph G = (V,E) and a set L of simple paths on G, called lines. The main task is to place the lines on G, so that the number of crossings among pairs of lines is minimized.

Our main contribution is two polynomial time algorithms. The first solves the general case of the MLCM problem, where the lines that traverse a particular vertex of G are allowed to use any side of it to either “enter” or “exit”, assuming that the endpoints of the lines are located at vertices of degree one. The second one solves more efficiently the restricted case, where only the left and the right side of each vertex can be used.

To the best of our knowledge, this is the first time where the general case of the MLCM problem is solved. Previous work was devoted to the restricted case of the MLCM problem under the additional assumption that the endpoints of the lines are either the topmost or the bottommost in their corresponding vertices, i.e., they are either on top or below the lines that pass through the vertex. Even for this case, we improve a known result of Asquith et al. from O(|E|5/2|L|3) to O(|V|(|E| + |L|)).

## Keywords

Polynomial Time Algorithm Terminal Station Simple Path Incoming Edge Underlying Network
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Authors and Affiliations

• Evmorfia Argyriou
• 1
• Michael A. Bekos
• 1
• Michael Kaufmann
• 2
• Antonios Symvonis
• 1
1. 1.School of Applied Mathematical & Physical SciencesNational Technical University of AthensGreece
2. 2.Institute for InformaticsUniversity of TübingenGermany