$\newcommand{\floor}[1]{\left\lfloor {#1} \right\rfloor} \renewcommand{\Re}{\mathbb{R}}$ 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.

翻译：$\newcommand{\floor}[1]{\left\lfloor {#1} \right\rfloor} \renewcommand{\Re}{\mathbb{R}}$ Tverberg定理指出，$\Re^d$中的$n$个点集可被划分为$\floor{n/(d+1)}$个具有公共交集的子集。该交集内的点（即Tverberg点）是输入点集的一个中心点，而Tverberg划分提供了这一性质的紧凑证明，具有重要的算法应用价值。然而，精确计算Tverberg点需要$n^{O(d^2)}$时间。本文针对该问题提出了若干新的近似算法，在运行时间或近似质量（或两者）上实现了改进。特别地，我们首次给出了一个强多项式（关于$n$和$d$）的近似算法用于寻找Tverberg点。