Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Smoothed Analysis of Order Types   Consider an ordered point set $P = (p_1,\ldots,p_n)$, its order type (denoted
by $\chi_P$) is a map which assigns to every triple of points a value in
$\{+,-,0\}$ based on whether the points are collinear(0), oriented clockwise(-)
or counter-clockwise(+). An abstract order type is a map $\chi :
\left[\substack{n\\3}\right] \rightarrow \{+,-,0\}$ (where
$\left[\substack{n\\3}\right]$ is the collection of all triples of a set of $n$
elements) that satisfies the following condition: for every set of five
elements $S\subset [n]$ its induced order type $\chi_{|S}$ is realizable by a
point set. To be precise, a point set $P$ realizes an order type $\chi$,if
$\chi_P(p_i,p_j,p_k) = \chi(i,j,k)$, for all $i<j<k$. Planar point sets are
among the most basic and natural geometric objects of study in Discrete and
Computational Geometry. Properties of point sets are relevant in theory and
practice alike. It is known, that deciding if an abstract order type is
realizable is complete for the existential theory of the reals. Our results
show that order type realizability is much easier for realistic instances than
in the worst case. In particular, we can recognize instances in "expected
\NP-time". This is one of the first $\exists\mathbb{R}$-complete problems
analyzed under the lens of Smoothed Analysis.

18
The categories of this paper are: econ
22
The categories of this paper are: chao-dyn
16
The categories of this paper are: stat
24
The categories of this paper are: dg-ga
12
The categories of this paper are: nucl-th
32
The categories of this paper are: atom-ph
25
The categories of this paper are: patt-sol
5
The categories of this paper are: gr-qc
15
The categories of this paper are: nucl-ex
7
The categories of this paper are: hep-th
33
The categories of this paper are: plasm-ph
10
The categories of this paper are: q-bio
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The categories of this paper are: chem-ph
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The categories of this paper are: adap-org
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The categories of this paper are: funct-an
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The categories of this paper are: mtrl-th