Complexity results on untangling red-blue matchings

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 $\left(\begin{array}{c}n\\ 2\end{array}\right)$ always exists, and (iii) the number of flips in any sequence never exceeds $\left(\begin{array}{c}n\\ 2\end{array}\right)\frac{n+4}{6}$. Finally, we present (iv) a lower bounding flip sequence with roughly $1.5\left(\begin{array}{c}n\\ 2\end{array}\right)$ flips, which shows that the $\left(\begin{array}{c}n\\ 2\end{array}\right)$ 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.