高维空间中最小曼哈顿网络的近似算法
Approximating Minimum Manhattan Networks in Higher Dimensions
AI总结：
针对高维最小曼哈顿网络问题三维尚无近似算法的现状，提出任意固定维下的$O(n^\eps)$近似算法，及三维端点在$k\ge2$个平行平面时的$4(k-1)$近似算法。
摘要：
我们研究最小曼哈顿网络问题，其定义如下：给定$\u211d^d$中一组被称为\textbf{端点（terminals）}的点，找到一个长度最小的网络，使得每对端点都由一组轴平行线段连接，且这些线段的总长度等于该端点对的曼哈顿（即$L_1$）距离。该问题在二维中是NP难的，在三维中不存在PTAS（除非${\cal P}={\cal NP}$）。目前已有针对二维的近似算法，但尚无三维的相关算法。\n 我们提出了适用于任意固定维度$d$和任意$\eps>0$的$O(n^\eps)$近似算法。针对三维场景，我们还给出了当端点包含在$k\ge2$个平行平面的并集中时的$4(k-1)$近似算法。
英文摘要：
We study the minimum Manhattan network problem, which is defined as follows. Given a set of points called \emph{terminals} in $\R^d$, find a minimum-length network such that each pair of terminals is connected by a set of axis-parallel line segments whose total length is equal to the pair's Manhattan (that is, $L_1$-) distance. The problem is NP-hard in 2D and there is no PTAS for 3D (unless ${\cal P}={\cal NP}$). Approximation algorithms are known for 2D, but not for 3D. We present, for any fixed dimension $d$ and any $\eps>0$, an $O(n^\eps)$-approximation algorithm. For 3D, we also give a $4(k-1)$-approximation algorithm for the case that the terminals are contained in the union of $k \ge 2$ parallel planes.
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。
