Complexity results on untangling red-blue matchings

Given a matching between n red points and n blue points by line segments in the plane, we consider the problem of obtaining a crossing-free matching through flip operations that replace two crossing segments by two non-crossing ones. We first show that (i) it is NP-hard to α-approximate the shortest...

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Bibliographic Details
Published inComputational geometry : theory and applications Vol. 111; p. 101974
Main Authors Das, Arun Kumar, Das, Sandip, da Fonseca, Guilherme D., Gerard, Yan, Rivier, Bastien
Format Journal Article
LanguageEnglish
Published Elsevier B.V 01.04.2023
Elsevier
Subjects
Computer Science
Matching
NP-hard
Planar geometry
Reconfiguration
Matching
NP-hard
Planar geometry
Reconfiguration
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Summary:Given a matching between n red points and n blue points by line segments in the plane, we consider the problem of obtaining a crossing-free matching through flip operations that replace two crossing segments by two non-crossing ones. We first show that (i) it is NP-hard to α-approximate the shortest flip sequence, for any constant α. Second, we show that when the red points are collinear, (ii) given a matching, a flip sequence of length at most (n2) always exists, and (iii) the number of flips in any sequence never exceeds (n2)n+46. Finally, we present (iv) a lower bounding flip sequence with roughly 1.5(n2) flips, which shows that the (n2) flips attained in the convex case are not the maximum, and (v) a convex matching from which any flip sequence has roughly 1.5n flips. The last four results, based on novel analyses, improve the constants of state-of-the-art bounds.
ISSN:0925-7721
DOI:10.1016/j.comgeo.2022.101974