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.

给定平面中的线段在 n 个红点和 n 个蓝点之间的匹配，我们考虑通过翻转操作将两个交叉线段替换为两个非交叉线段来获得无交叉匹配的问题。我们首先证明 (i) 对于任何常量α ， α逼近最短翻转序列是 NP 难的。其次，我们证明当红点共线时，(ii) 给定一个匹配，翻转序列的长度最多$\left(\begin{array}{c}n\\ \mathrm{2个}\end{array}\right)$总是存在的，并且（iii）任何序列中的翻转次数永远不会超过$\left(\begin{array}{c}n\\ \mathrm{2个}\end{array}\right)\frac{n+\mathrm{4个}}{\mathrm{6个}}$. 最后，我们提出（iv）一个下边界翻转序列，大致$1.5\left(\begin{array}{c}n\\ \mathrm{2个}\end{array}\right)$翻转，这表明$\left(\begin{array}{c}n\\ \mathrm{2个}\end{array}\right)$在凸情况下获得的翻转不是最大值，并且 (v) 一个凸匹配，从中任何翻转序列都大致具有$1.5n$翻转。最后四个结果基于新颖的分析，改进了最先进边界的常数。