Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Rainbow at the End of the Line --- A PPAD Formulation of the
  Colorful Carath\'eodory Theorem with Applications   Let $C_1,...,C_{d+1}$ be $d+1$ point sets in $\mathbb{R}^d$, each containing
the origin in its convex hull. A subset $C$ of $\bigcup_{i=1}^{d+1} C_i$ is
called a colorful choice (or rainbow) for $C_1, \dots, C_{d+1}$, if it contains
exactly one point from each set $C_i$. The colorful Carath\'eodory theorem
states that there always exists a colorful choice for $C_1,\dots,C_{d+1}$ that
has the origin in its convex hull. This theorem is very general and can be used
to prove several other existence theorems in high-dimensional discrete
geometry, such as the centerpoint theorem or Tverberg's theorem. The colorful
Carath\'eodory problem (CCP) is the computational problem of finding such a
colorful choice. Despite several efforts in the past, the computational
complexity of CCP in arbitrary dimension is still open.
  We show that CCP lies in the intersection of the complexity classes PPAD and
PLS. This makes it one of the few geometric problems in PPAD and PLS that are
not known to be solvable in polynomial time. Moreover, it implies that the
problem of computing centerpoints, computing Tverberg partitions, and computing
points with large simplicial depth is contained in $\text{PPAD} \cap
\text{PLS}$. This is the first nontrivial upper bound on the complexity of
these problems. Finally, we show that our PPAD formulation leads to a
polynomial-time algorithm for a special case of CCP in which we have only two
color classes $C_1$ and $C_2$ in $d$ dimensions, each with the origin in its
convex hull, and we would like to find a set with half the points from each
color class that contains the origin in its convex hull.

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The categories of this paper are: hep-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: atom-ph
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The categories of this paper are: math
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The categories of this paper are: hep-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: eess
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The categories of this paper are: cond-mat
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The categories of this paper are: q-fin
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-lat
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The categories of this paper are: hep-th
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The categories of this paper are: solv-int
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-alg
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The categories of this paper are: chem-ph