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 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 shows that the n(n-1)/2 flips attained in the convex case are not the maximum, and (v) a convex matching from which any flip sequence has roughly 1.5 n flips. The last four results, based on novel analyses, improve the constants of state-of-the-art bounds.
翻译:鉴于正红点和正蓝点之间在平面线段之间的匹配,我们考虑通过翻转操作获得无交叉匹配的问题,翻转操作以两个非交叉部分取代两个跨段。我们首先显示:(一) 任何恒定的阿尔法,它硬至最短的翻转序列。第二,我们显示,当红点为线性线性,(二) 给定一个匹配,最大为n(n-1/2)/2的翻转序列始终存在,以及(三) 任何序列的翻转次数从未超过(n(n)(n)(l/2)(n+4)/6)。最后,我们提出(四) 下一个下拉链性翻转序列,大约为1.5n(n)(n)(n)(l/2)/2),这显示在 convex案中达到的n(n)(n)(m)/2翻转码不是最大,以及(五) 螺旋相匹配,任何翻转序列都大约为1.5n(n)(n)(n)(n)(n)(n)(n)(n)(n)(l+4)/)//6。最后,根据新分析,我们提出(d),我们提出(四),这四个结果,可以改进了状态框的常数。