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Journal Articles Computational Geometry Year : 2023

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 alpha-approximate the shortest flip sequence, for any constant alpha. Second, we show that when the red points are collinear, (ii) given a matching, a flip sequence of length at most $\binom{n}{2}$ always exists, and (iii) the number of flips in any sequence never exceeds $\binom{n}{2}\frac{n+4}{6}$. Finally, we present (iv) a lower bounding flip sequence with roughly $1.5 \binom{n}{2}$ flips, which shows that the $\binom{n}{2}$ 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.

Dates and versions

hal-04072822 , version 1 (18-04-2023)



Arun Kumar Das, Sandip Das, Guilherme da Fonseca, Yan Gerard, Bastien Rivier. Complexity results on untangling red-blue matchings. Computational Geometry, 2023, 111, pp.101974. ⟨10.1016/j.comgeo.2022.101974⟩. ⟨hal-04072822⟩
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