Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Finding Planted Cliques in Sublinear Time   We study the planted clique problem in which a clique of size $k$ is planted
in an Erd\H{o}s-R\'enyi graph of size $n$ and one wants to recover this planted
clique. For $k=\Omega(\sqrt{n})$, polynomial time algorithms can find the
planted clique. The fastest such algorithms run in time linear $O(n^2)$ (or
nearly linear) in the size of the input [FR10,DGGP14,DM15a]. In this work, we
initiate the development of sublinear time algorithms that find the planted
clique when $k=\omega(\sqrt{n \log \log n})$. Our algorithms can recover the
clique in time
$\widetilde{O}\left(n+(\frac{n}{k})^{3}\right)=\widetilde{O}\left(n^{\frac{3}{2}}\right)$
when $k=\Omega(\sqrt{n\log n})$, and in time
$\widetilde{O}\left(n^2/\exp{\left(\frac{k^2}{24n}\right)}\right)$ for
$\omega(\sqrt{n\log \log n})=k=o(\sqrt{n\log{n}})$. An ${\Omega}(n)$ running
time lower bound for the planted clique recovery problem follows easily from
the results of [RS19] and therefore our recovery algorithms are optimal
whenever $k = \Omega(n^{\frac{2}{3}})$. As the lower bound of [RS19] builds on
purely information theoretic arguments, it cannot provide a detection lower
bound stronger than $\widetilde{\Omega}(\frac{n^2}{k^2})$. Since our algorithms
for $k = \Omega(\sqrt{n \log n})$ run in time
$\widetilde{O}\left(\frac{n^3}{k^3} + n\right)$, we show stronger lower bounds
based on computational hardness assumptions. With a slightly different notion
of the planted clique problem we show that the Planted Clique Conjecture
implies the following. A natural family of non-adaptive algorithms---which
includes our algorithms for clique detection---cannot reliably solve the
planted clique detection problem in time $O\left(
\frac{n^{3-\delta}}{k^3}\right)$ for any constant $\delta>0$. Thus we provide
evidence that if detecting small cliques is hard, it is also likely that
detecting large cliques is not \textit{too} easy.

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The categories of this paper are: comp-gas
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The categories of this paper are: stat
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The categories of this paper are: chem-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-alg
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The categories of this paper are: acc-phys
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The categories of this paper are: supr-con
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The categories of this paper are: astro-ph
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The categories of this paper are: physics
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-ph
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The categories of this paper are: q-bio
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The categories of this paper are: nucl-th
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The categories of this paper are: econ
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The categories of this paper are: plasm-ph
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The categories of this paper are: adap-org