Complexity results on untangling red-blue matchings

Article Type

Research Article

Publication Title

Computational Geometry: Theory and Applications

Abstract

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 [Formula presented]. 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.

DOI

https://10.1016/j.comgeo.2022.101974

Publication Date

4-1-2023

Comments

Open Access, Bronze

This document is currently not available here.

Share

COinS