Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Polynomial Turing Kernels for Clique with an Optimal Number of Queries   A polynomial Turing kernel for some parameterized problem $P$ is a
polynomial-time algorithm that solves $P$ using queries to an oracle of $P$
whose sizes are upper-bounded by some polynomial in the parameter. Here the
term "polynomial" refers to the bound on the query sizes, as the running time
of any kernel is required to be polynomial. One of the most important open
goals in parameterized complexity is to understand the applicability and
limitations of polynomial Turing Kernels. As any fixed-parameter tractable
problem admits a Turing kernel of some size, the focus has mostly being on
determining which problems admit such kernels whose query sizes can be indeed
bounded by some polynomial.
  In this paper we take a different approach, and instead focus on the number
of queries that a Turing kernel uses, assuming it is restricted to using only
polynomial sized queries. Our study focuses on one the main problems studied in
parameterized complexity, the Clique problem: Given a graph $G$ and an integer
$k$, determine whether there are $k$ pairwise adjacent vertices in $G$. We show
that Clique parameterized by several structural parameters exhibits the
following phenomena:
  - It admits polynomial Turing kernels which use a sublinear number of
queries, namely $O(n/\log^c n)$ queries where $n$ is the total size of the
graph and $c$ is any constant. This holds even for a very restrictive type of
Turing kernels which we call OR-kernels.
  - It does not admit polynomial Turing kernels which use $O(n^{1-\epsilon})$
queries, unless NP$\subseteq$coNP/poly.
  For proving the second item above, we develop a new framework for bounding
the number of queries needed by polynomial Turing kernels. This framework is
inspired by the standard lower bounds framework for Karp kernels, and while it
is quite similar, it still requires some novel ideas to allow its extension to
the Turing setting.

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The categories of this paper are: math-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: comp-gas
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The categories of this paper are: chao-dyn
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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: chem-ph
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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: quant-ph
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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: hep-ex
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The categories of this paper are: nlin
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The categories of this paper are: alg-geom
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The categories of this paper are: solv-int