Journey to the Center of the Point Set

We revisit an algorithm of Clarkson et al. [K. L. Clarkson et al., 1996], that computes (roughly) a 1/(4d^2)-centerpoint in O~(d^9) time, for a point set in R^d, where O~ hides polylogarithmic terms. We present an improved algorithm that computes (roughly) a 1/d^2-centerpoint with running time O~(d^7). While the improvements are (arguably) mild, it is the first progress on this well known problem in over twenty years. The new algorithm is simpler, and the running time bound follows by a simple random walk argument, which we believe to be of independent interest. We also present several new applications of the improved centerpoint algorithm

No-dimensional Tverberg Partitions Revisited

\newcommand{\epsA}{\Mh{\delta}} \newcommand{\Re}{\mathbb{R}} \newcommand{\reals}{\mathbb{R}} \newcommand{\SetX}{\mathsf{X}} \newcommand{\diam}{\Delta} \newcommand{\Mh}[1]{#1} \newcommand{\query}{q} \newcommand{\eps}{\varepsilon} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{\IntRange}[1]{[ #1 ]} \newcommand{\Space}{\overline{\mathsf{m}}} \newcommand{\pth}[2][\!]{#1\left({#2}\right)} \newcommand{\polylog}{\mathrm{polylog}} \newcommand{\N}{\mathbb N} \newcommand{\Z}{\mathbb Z} \newcommand{\pt}{p} \newcommand{\distY}[2]{\left\| {#1} - {#2} \right\|} \newcommand{\PP}{P} \newcommand{\ptq}{q} \newcommand{\pts}{s} Given a set \PP \subset \Re^d of $n$ points, with diameter \diam, and a parameter \epsA \in (0,1), it is known that there is a partition of \PP into sets \PP_1, \ldots, \PP_t, each of size O(1/\epsA^2), such that their convex-hulls all intersect a common ball of radius \epsA \diam. We prove that a random partition, with a simple alteration step, yields the desired partition, resulting in a linear time algorithm. Previous proofs were either existential (i.e., at least exponential time), or required much bigger sets. In addition, the algorithm and its proof of correctness are significantly simpler than previous work, and the constants are slightly better. In addition, we provide a linear time algorithm for computing a ``fuzzy'' centerpoint. We also prove a no-dimensional weak \eps-net theorem with an improved constant

On Undecided LP, Clustering and Active Learning

We study colored coverage and clustering problems. Here, we are given a colored point set, where the points are covered by k (unknown) clusters, which are monochromatic (i.e., all the points covered by the same cluster have the same color). The access to the colors of the points (or even the points themselves) is provided indirectly via various oracle queries (such as nearest neighbor, or separation queries). We show that one can correctly deduce the color of all the points (i.e., compute a monochromatic clustering of the points) using a polylogarithmic number of queries, if the number of clusters is a constant. We investigate several variants of this problem, including Undecided Linear Programming and covering of points by k monochromatic balls

Improved Approximation Algorithms for Tverberg Partitions

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

No-Dimensional Tverberg Theorems and Algorithms

Tverberg's theorem is a classic result in discrete geometry. It states that for any integer $k \ge 2$ and any finite $d$-dimensional point set $P \subseteq \mathbb{R}^d$ of at least $(d + 1)(k - 1) + 1$ points, we can partition $P$ into $k$ subsets whose convex hulls have a non-empty intersection. The computational problem of finding such a partition lies in the complexity class $\mathrm{PPAD} \cap \mathrm{PLS}$, but no hardness results are known. Tverberg's theorem also has a colorful variant: the points in $P$ have colors, and under certain conditions, $P$ can be partitioned into colorful sets, i.e., sets in which each color appears exactly once such that the convex hulls of the sets intersect. Recently, Adiprasito, Barany, and Mustafa [SODA 2019] proved a no-dimensional version of Tverberg's theorem, in which the convex hulls of the sets in the partition may intersect in an approximate fashion, relaxing the requirement on the cardinality of $P$. The argument is constructive, but it does not result in a polynomial-time algorithm. We present an alternative proof for a no-dimensional Tverberg theorem that leads to an efficient algorithm to find the partition. More specifically, we show an deterministic algorithm that finds for any set $P \subseteq \mathbb{R}^d$ of $n$ points and any $k \in \{2, \dots, n\}$ in $O(nd \log k )$ time a partition of $P$ into $k$ subsets such that there is a ball of radius $O\left(\frac{k}{\sqrt{n}}\textrm{diam}(P)\right)$ intersecting the convex hull of each subset. A similar result holds also for the colorful version. To obtain our result, we generalize Sarkaria's tensor product constructions [Israel Journal Math., 1992] that reduces the Tverberg problem to the Colorful Caratheodory problem. By carefully choosing the vectors used in the tensor products, we implement the reduction in an efficient manner.Comment: A shorter version will appear at SoCG 202

No-Dimensional Tverberg Theorems and Algorithms

Tverberg’s theorem states that for any k≥2 and any set P⊂Rd 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 CLS=PPAD∩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, Bárány, and Mustafa (SODA 2019) 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⊂Rd and any k∈{2,…,n} in O(nd⌈logk⌉) time a k-partition of P such that there is a ball of radius O((k/n−−√)diam(P)) that intersects the convex hull of each set. Given that this problem is not known to be solvable exactly in polynomial time, our result provides a remarkably efficient and simple new notion of approximation. Our main contribution is to generalize Sarkaria’s method (Israel Journal Math., 1992) to reduce the Tverberg problem to the colorful Carathéodory problem (in the simplified tensor product interpretation of Bárány and Onn) and to apply it algorithmically. It turns out that this not only leads to an alternative algorithmic proof of a no-dimensional Tverberg theorem, but it also generalizes to other settings such as the colorful variant of the problem