Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Promise Constraint Satisfaction and Width   We study the power of the bounded-width consistency algorithm in the context
of the fixed-template Promise Constraint Satisfaction Problem (PCSP). Our main
technical finding is that the template of every PCSP that is solvable in
bounded width satisfies a certain structural condition implying that its
algebraic closure-properties include weak near unanimity polymorphisms of all
large arities. While this parallels the standard (non-promise) CSP theory, the
method of proof is quite different and applies even to the regime of sublinear
width. We also show that, in contrast with the CSP world, the presence of weak
near unanimity polymorphisms of all large arities does not guarantee
solvability in bounded width. The separating example is even solvable in the
second level of the Sherali-Adams (SA) hierarchy of linear programming
relaxations. This shows that, unlike for CSPs, linear programming can be
stronger than bounded width. A direct application of these methods also show
that the problem of $q$-coloring $p$-colorable graphs is not solvable in
bounded or even sublinear width, for any two constants $p$ and $q$ such that $3
\leq p \leq q$. Turning to algorithms, we note that Wigderson's algorithm for
$O(\sqrt{n})$-coloring $3$-colorable graphs with $n$ vertices is implementable
in width $4$. Indeed, by generalizing the method we see that, for any $\epsilon
> 0$ smaller than $1/2$, the optimal width for solving the problem of
$O(n^\epsilon)$-coloring $3$-colorable graphs with $n$ vertices lies between
$n^{1-3\epsilon}$ and $n^{1-2\epsilon}$. The upper bound gives a simple
$2^{\Theta(n^{1-2\epsilon}\log(n))}$-time algorithm that, asymptotically, beats
the straightforward $2^{\Theta(n^{1-\epsilon})}$ bound that follows from
partitioning the graph into $O(n^\epsilon)$ many independent parts each of size
$O(n^{1-\epsilon})$.

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The categories of this paper are: chem-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: q-bio
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The categories of this paper are: eess
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The categories of this paper are: chao-dyn
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The categories of this paper are: cond-mat
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The categories of this paper are: ao-sci
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The categories of this paper are: quant-ph
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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: acc-phys
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-ex
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The categories of this paper are: bayes-an
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The categories of this paper are: adap-org