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Complexity Results on Untangling Planar Rectilinear Red-Blue Matchings

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Abstract

Given a rectilinear matching between n red points and n blue points 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 colinear, (ii) given a matching, a flip sequence of length at most n(n-1)/2 always exists and (iii) the number of flips in any sequence never exceeds (n(n-1)/2) (n+4)/6. Finally, we present (iv) a lower bounding flip sequence with roughly 1.5 n(n-1)/2 flips, which disproves the conjecture that n(n-1)/2, reached in the convex case, is the maximum. The last three results, based on novel analyses, improve the constants of state-of-the-art bounds.

Dates and versions

hal-03588674 , version 1 (25-02-2022)

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Arun Kumar Das, Sandip Das, Guilherme D. da Fonseca, Yan Gerard, Bastien Rivier. Complexity Results on Untangling Planar Rectilinear Red-Blue Matchings. 2022. ⟨hal-03588674⟩
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