Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Query Complexity of Local Search and Brouwer in Rounds   We consider the query complexity of finding a local minimum of a function
defined on a graph, where at most $k$ rounds of interaction with the oracle are
allowed. Rounds model parallel settings, where each query takes resources to
complete and is executed on a separate processor. Thus the query complexity in
$k$ rounds informs how many processors are needed to achieve a parallel time of
$k$.
  We focus on the d-dimensional grid $[n]^d$, where the dimension $d$ is a
constant, and consider two regimes for the number of rounds: constant and
polynomial in n. We give algorithms and lower bounds that characterize the
trade-off between the number of rounds of adaptivity and the query complexity
of local search.
  When the number of rounds $k$ is constant, we show that the query complexity
of local search in $k$ rounds is $\Theta\bigl(n^{\frac{d^{k+1} - d^k}{d^k -
1}}\bigl)$, for both deterministic and randomized algorithms. When the number
of rounds is polynomial, i.e. $k = n^{\alpha}$ for $0 < \alpha < d/2$, the
randomized query complexity is $\Theta\left(n^{d-1 -
\frac{d-2}{d}\alpha}\right)$ for all $d \geq 5$. For $d=3$ and $d=4$, we show
the same upper bound expression holds and give almost matching lower bounds.
The local search analysis also enables us to characterize the query complexity
of computing a Brouwer fixed point in rounds. Our proof technique for lower
bounding the query complexity in rounds may be of independent interest as an
alternative to the classical relational adversary method of Aaronson from the
fully adaptive setting.

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The categories of this paper are: bayes-an
33
The categories of this paper are: plasm-ph
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The categories of this paper are: q-alg
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The categories of this paper are: adap-org
5
The categories of this paper are: gr-qc
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The categories of this paper are: math-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: atom-ph
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The categories of this paper are: physics
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The categories of this paper are: nucl-th
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The categories of this paper are: supr-con
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The categories of this paper are: comp-gas
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The categories of this paper are: hep-th
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The categories of this paper are: nucl-ex
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The categories of this paper are: eess
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The categories of this paper are: alg-geom