Algorithms for Tolerant Tverberg Partitions

TL;DR

This paper studies algorithms for finding Tverberg partitions that are tolerant to point deletions, providing polynomial-time solutions for low dimensions, an approximation algorithm for higher dimensions, and complexity results.

Contribution

It improves bounds and algorithms for t-tolerant Tverberg partitions, including polynomial-time methods in low dimensions and an approximation approach in higher dimensions.

Findings

Polynomial-time algorithms for $d \,\leq\, 2$

First approximation algorithm for $d \,\geq\, 3$

T- tolerance decision problem is coNP-complete

Abstract

Let $P$ be a $d$-dimensional $n$-point set. A partition $T$ of $P$ is called a Tverberg partition if the convex hulls of all sets in $T$ intersect in at least one point. We say $T$ is $t$-tolerant if it remains a Tverberg partition after deleting any $t$ points from $P$. Sober\'{o}n and Strausz proved that there is always a $t$-tolerant Tverberg partition with $\lceil n / (d+1)(t+1) \rceil$ sets. However, so far no nontrivial algorithms for computing or approximating such partitions have been presented. For $d \leq 2$, we show that the Sober\'{o}n-Strausz bound can be improved, and we show how the corresponding partitions can be found in polynomial time. For $d \geq 3$, we give the first polynomial-time approximation algorithm by presenting a reduction to the Tverberg problem with no tolerance. Finally, we show that it is coNP-complete to determine whether a given Tverberg partition is t-tolerant.