平面双色瓶颈生成树
Planar Bichromatic Bottleneck Spanning Trees
摘要：
给定平面中$n$红色和蓝色点的集合$P$，$P$的\emph{planar bichromatic spanning tree}是$P$的生成树，使得每条边连接在红色和蓝色点之间，并且没有两条边相交。在瓶颈平面双色生成树问题中，目标是找到一棵平面双色生成树$T$，使得$T$中最长边的长度最小。在本文中，我们证明了这个问题是NP-困难的一般位置的点。通过证明瓶颈$λ$的任意双色生成树最多可以转化为瓶颈的平面双色生成树$8\sqrt{2}λ$，给出了一个多项式时间的$(8\sqrt{2})$ -近似算法.
英文摘要：
Given a set $P$ of $n$ red and blue points in the plane, a \emph{planar bichromatic spanning tree} of $P$ is a spanning tree of $P$, such that each edge connects between a red and a blue point, and no two edges intersect. In the bottleneck planar bichromatic spanning tree problem, the goal is to find a planar bichromatic spanning tree $T$, such that the length of the longest edge in $T$ is minimized. In this paper, we show that this problem is NP-hard for points in general position. Moreover, we present a polynomial-time $(8\sqrt{2})$-approximation algorithm, by showing that any bichromatic spanning tree of bottleneck $λ$ can be converted to a planar bichromatic spanning tree of bottleneck at most $8\sqrt{2}λ$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
