Algorithms for Tolerated Tverberg Partitions

Abstract

Let P be a d-dimensional n-point set. A partition $\mathcal{T}$ of P is called a Tverberg partition if the convex hulls of all sets in $\mathcal{T}$ intersect in at least one point. We say $\mathcal{T}$ is t-tolerated if it remains a Tverberg partition after deleting any t points from P. Soberón and Strausz proved that there is always a t-tolerated Tverberg partition with ⌈n / (d + 1)(t + 1) ⌉ sets. However, so far no nontrivial algorithms for computing or approximating such partitions have been presented.

For d ≤ 2, we show that the Soberón-Strausz bound can be improved, and we show how the corresponding partitions can be found in polynomial time. For d ≥ 3, we give the first polynomial-time approximation algorithm by presenting a reduction to the (untolerated) Tverberg problem. Finally, we show that it is coNP-complete to determine whether a given Tverberg partition is t-tolerated.