Improved Approximation Algorithms for Tverberg Partitions

TL;DR

This paper introduces new approximation algorithms for Tverberg partitions that improve computational efficiency and approximation quality, including the first strongly polynomial algorithm for finding a Tverberg point.

Contribution

The paper presents the first strongly polynomial approximation algorithm for Tverberg points, improving both speed and accuracy over previous methods.

Findings

First strongly polynomial approximation algorithm for Tverberg points

Improved running time for computing approximate Tverberg points

Enhanced approximation quality in Tverberg partition algorithms

Abstract

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