Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A logic-based algorithmic meta-theorem for mim-width   We introduce a logic called distance neighborhood logic with acyclicity and
connectivity constraints ($\mathsf{A\&C~DN}$ for short) which extends
existential $\mathsf{MSO_1}$ with predicates for querying neighborhoods of
vertex sets and for verifying connectivity and acyclicity of vertex sets in
various powers of a graph. Building upon [Bergougnoux and Kant\'e, ESA 2019;
SIDMA 2021], we show that the model checking problem for every fixed
$\mathsf{A\&C~DN}$ formula is solvable in $n^{O(w)}$ time when the input graph
is given together with a branch decomposition of mim-width $w$. Nearly all
problems that are known to be solvable in polynomial time given a branch
decomposition of constant mim-width can be expressed in this framework. We add
several natural problems to this list, including problems asking for diverse
sets of solutions. Our model checking algorithm is efficient whenever the given
branch decomposition of the input graph has small index in terms of the
$d$-neighborhood equivalence [Bui-Xuan, Telle, and Vatshelle, TCS 2013]. We
therefore unify and extend known algorithms for tree-width, clique-width and
rank-width. Our algorithm has a single-exponential dependence on these three
width measures and asymptotically matches run times of the fastest known
algorithms for several problems. This results in algorithms with tight run
times under the Exponential Time Hypothesis ($\mathsf{ETH}$) for tree-width and
clique-width; the above mentioned run time for mim-width is nearly tight under
the $\mathsf{ETH}$ for several problems as well. Our results are also tight in
terms of the expressive power of the logic: we show that already slight
extensions of our logic make the model checking problem para-$\mathsf{NP}$-hard
when parameterized by mim-width plus formula length.

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The categories of this paper are: nucl-ex
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The categories of this paper are: nucl-th
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The categories of this paper are: adap-org
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The categories of this paper are: math
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The categories of this paper are: funct-an
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The categories of this paper are: eess
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The categories of this paper are: plasm-ph
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The categories of this paper are: hep-th
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The categories of this paper are: cond-mat
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The categories of this paper are: alg-geom
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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: quant-ph
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The categories of this paper are: solv-int
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The categories of this paper are: hep-ex
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The categories of this paper are: q-alg