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No-dimensional Tverberg Theorems and Algorithms by Choudhary, Aruni; Mulzer, Wolfgang is a scholarly article available to read on EtoBox.

What is No-dimensional Tverberg Theorems and Algorithms about?

Tverberg's theorem states that for any $k \ge 2$ and any set $P \subset \mathbb{R}^d$ of at least $(d + 1)(k - 1) + 1$ points in $d$ dimensions, we can partition $P$ into $k$ subsets whose convex hulls have a non-empty intersection. The associated search problem of finding the partition lies in the complexity class $\text{CLS} = \text{PPAD} \cap \text{PLS}$, but no hardness results are known. In the colorful Tverberg theorem, the points in $P$ have colors, and under certain conditions, $P$ can be partitioned into colorful sets, in which each color appears exactly once and whose convex hulls intersect. To date, the complexity of the associated search problem is unresolved. Recently, Adiprasito, Barany, and Mustafa gave a no-dimensional Tverberg theorem, in which the convex hulls may intersect in an approximate fashion. This relaxes the requirement on the cardinality of $P$. The argument is constructive, but does not result in a polynomial-time algorithm. We present a deterministic algorithm that finds for any $n$-point set $P \subset \mathbb{R}^d$ and any $k \in \{2, \dots, n\}$ in $O(nd \lceil{\log k}\rceil)$ time a $k$-partition of $P$ such that there is a ball of radius $O\left((

Author

Choudhary, Aruni; Mulzer, Wolfgang

Published

2019

Language

EN