Ordering Metro Lines by Block Crossings

TL;DR

This paper addresses the problem of reducing crossings in transportation network drawings by merging them into block crossings, providing algorithms with bounds for special graph classes and general graphs.

Contribution

It introduces approximation algorithms for minimizing block crossings, including an asymptotically optimal algorithm for general graphs, and proves NP-hardness of the problem.

Findings

Provided approximation algorithms with bounds for special classes of graphs.

Developed an asymptotically worst-case optimal algorithm for general graphs.

Proved the NP-hardness of minimizing block crossings.

Abstract

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 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. That is, we bound the number of block crossings that our algorithm needs and construct worst-case instances on which the number of block crossings that is necessary in any solution is asymptotically the same as our bound.