Exact and fixed-parameter algorithms for metro-line crossing minimization problems

TL;DR

This paper introduces new exact and fixed-parameter algorithms for minimizing line crossings in metro-line drawings, including polynomial-time and FPT solutions for specific variants of the problem.

Contribution

The paper presents the first polynomial-time algorithm for crossing-free line drawings and develops a fixed-parameter tractable algorithm based on line multiplicity.

Findings

Deciding crossing-free drawings is polynomial-time solvable.

A fast exponential algorithm for crossing minimization is proposed.

The problem is fixed-parameter tractable with respect to line multiplicity.

Abstract

A metro-line crossing minimization problem is to draw multiple lines on an underlying graph that models stations and rail tracks so that the number of crossings of lines becomes minimum. It has several variations by adding restrictions on how lines are drawn. Among those, there is one with a restriction that line terminals have to be drawn at a verge of a station, and it is known to be NP-hard even when underlying graphs are paths. This paper studies the problem in this setting, and propose new exact algorithms. We first show that a problem to decide if lines can be drawn without crossings is solved in polynomial time, and propose a fast exponential algorithm to solve a crossing minimization problem. We then propose a fixed-parameter algorithm with respect to the multiplicity of lines, which implies that the problem is FPT.