Metro-Line Crossing Minimization: Hardness, Approximations, and Tractable Cases

TL;DR

This paper proves that the metro-line crossing minimization problem is NP-hard and provides an approximation algorithm for a related variant, advancing understanding of its computational complexity and solutions.

Contribution

It establishes the NP-hardness of MLCM and introduces an $O(\sqrt{\log |L|})$-approximation algorithm for MLCM-P, a variant with line end placement constraints.

Findings

MLCM is NP-hard.

An approximation algorithm with $O(\sqrt{\log |L|})$ ratio is proposed for MLCM-P.

The results resolve open questions about the problem's complexity and approximability.

Abstract

Crossing minimization is one of the central problems in graph drawing. Recently, there has been an increased interest in the problem of minimizing crossings between paths in drawings of graphs. This is the metro-line crossing minimization problem (MLCM): Given an embedded graph and a set L of simple paths, called lines, order the lines on each edge so that the total number of crossings is minimized. So far, the complexity of MLCM has been an open problem. In contrast, the problem variant in which line ends must be placed in outermost position on their edges (MLCM-P) is known to be NP-hard. Our main results answer two open questions: (i) We show that MLCM is NP-hard. (ii) We give an $O(\sqrt{\log |L|})$-approximation algorithm for MLCM-P.