GD 2013: Graph Drawing pp 328-339

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

• Martin Fink
• Sergey Pupyrev
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8242)

## 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.

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© Springer International Publishing Switzerland 2013

## Authors and Affiliations

• Martin Fink
• 1
• Sergey Pupyrev
• 2
• 3
1. 1.Lehrstuhl für Informatik IUniversität WürzburgGermany
2. 2.Department of Computer ScienceUniversity of ArizonaUSA
3. 3.Institute of Mathematics and Computer ScienceUral Federal UniversityRussia