Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: gr-qc
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The categories of this paper are: alg-geom
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The categories of this paper are: nlin
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The categories of this paper are: cmp-lg
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The categories of this paper are: adap-org
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The categories of this paper are: bayes-an
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The categories of this paper are: q-alg
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The categories of this paper are: ao-sci
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The categories of this paper are: cond-mat
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The categories of this paper are: quant-ph
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The categories of this paper are: eess
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The categories of this paper are: hep-ex
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The categories of this paper are: solv-int
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The categories of this paper are: hep-lat
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The categories of this paper are: acc-phys
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The categories of this paper are: physics