Tverberg划分的改进近似算法
Improved Approximation Algorithms for Tverberg Partitions
AI总结：
针对精确计算Tverberg点时间复杂度高的问题，提出若干改进的近似算法，提升运行时间与近似质量，还给出首个在n和d上均为强多项式的Tverberg点近似算法。
摘要：
Tverberg定理指出，$\u211d$中的n个点的集合可被划分为$\floor{n/(d+1)}$个具有公共交集的集合。该交集中的一个点（又称Tverberg点）是输入点集的中心点，且Tverberg划分提供了该结论的紧凑证明，具有算法应用价值。遗憾的是，精确计算Tverberg点需要$n^{O(d^2)}$时间。我们针对该问题提出了若干新的近似算法，在运行时间、近似质量或两者兼具方面均有提升。特别地，我们提出了首个在n和d上均为强多项式的Tverberg点求解近似算法。
英文摘要：
$\newcommand{\floor}[1]{\left\lfloor {#1} \right\rfloor} \renewcommand{\Re}{\mathbb{R}}$ Tverberg's theorem states that a set of $n$ points in $\Re^d$ can be partitioned into $\floor{n/(d+1)}$ sets with a common intersection. A point in this intersection (aka Tverberg point) is a centerpoint of the input point set, and the Tverberg partition provides a compact proof of this, which is algorithmically useful. Unfortunately, computing a Tverberg point exactly requires $n^{O(d^2)}$ time. We provide several new approximation algorithms for this problem, which improve either the running time or quality of approximation, or both. In particular, we provide the first strongly polynomial (in both $n$ and $d$) approximation algorithm for finding a Tverberg point.
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。
