Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Fully-Dynamic Coresets   With input sizes becoming massive, coresets -- small yet representative
summary of the input -- are relevant more than ever. A weighted set $C_w$ that
is a subset of the input is an $\varepsilon$-coreset if the cost of any
feasible solution $S$ with respect to $C_w$ is within $[1 {\pm} \varepsilon]$
of the cost of $S$ with respect to the original input. We give a very general
technique to compute coresets in the fully-dynamic setting where input points
can be added or deleted. Given a static $\varepsilon$-coreset algorithm that
runs in time $t(n, \varepsilon, \lambda)$ and computes a coreset of size $s(n,
\varepsilon, \lambda)$, where $n$ is the number of input points and $1
{-}\lambda$ is the success probability, we give a fully-dynamic algorithm that
computes an $\varepsilon$-coreset with worst-case update time $O((\log n) \cdot
t(s(n, \varepsilon/\log n, \lambda/n), \varepsilon/\log n, \lambda/n) )$ (this
bound is stated informally), where the success probability is $1{-}\lambda$.
Our technique is a fully-dynamic analog of the merge-and-reduce technique that
applies to insertion-only setting. Although our space usage is $O(n)$, we work
in the presence of an adaptive adversary, and we show that $\Omega(n)$ space is
required when adversary is adaptive. As a consequence, we get fully-dynamic
$\varepsilon$-coreset algorithms for $k$-median and $k$-means with worst-case
update time $O(\varepsilon^{-2}k^2\log^5 n \log^3 k)$ and coreset size
$O(\varepsilon^{-2}k\log n \log^2 k)$ ignoring $\log \log n$ and
$\log(1/\varepsilon)$ factors and assuming that $\varepsilon, \lambda =
\Omega(1/$poly$(n))$. These are the first fully-dynamic algorithms for
$k$-median and $k$-means with worst-case update times $O($poly$(k, \log n,
\varepsilon^{-1}))$. We also give conditional lower bound on update/query time
for any fully-dynamic $(4 - \delta)$-approximation algorithm for $k$-means.

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The categories of this paper are: q-fin
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The categories of this paper are: alg-geom
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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: ao-sci
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The categories of this paper are: dg-ga
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The categories of this paper are: physics
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The categories of this paper are: chem-ph
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The categories of this paper are: solv-int
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The categories of this paper are: math-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-alg
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The categories of this paper are: q-bio
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The categories of this paper are: acc-phys
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The categories of this paper are: cmp-lg