Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Lower Bounds on the Complexity of MSO1 Model-Checking   One of the most important algorithmic meta-theorems is a famous result by
Courcelle, which states that any graph problem definable in monadic
second-order logic with edge-set quantifications (i.e., MSO2 model-checking) is
decidable in linear time on any class of graphs of bounded tree-width.
Recently, Kreutzer and Tazari proved a corresponding complexity lower-bound -
that MSO2 model-checking is not even in XP wrt. the formula size as parameter
for graph classes that are subgraph-closed and whose tree-width is
poly-logarithmically unbounded. Of course, this is not an unconditional result
but holds modulo a certain complexity-theoretic assumption, namely, the
Exponential Time Hypothesis (ETH).
  In this paper we present a closely related result. We show that even MSO1
model-checking with a fixed set of vertex labels, but without edge-set
quantifications, is not in XP wrt. the formula size as parameter for graph
classes which are subgraph-closed and whose tree-width is poly-logarithmically
unbounded unless the non-uniform ETH fails. In comparison to Kreutzer and
Tazari; $(1)$ we use a stronger prerequisite, namely non-uniform instead of
uniform ETH, to avoid the effectiveness assumption and the construction of
certain obstructions used in their proofs; and $(2)$ we assume a different set
of problems to be efficiently decidable, namely MSO1-definable properties on
vertex labeled graphs instead of MSO2-definable properties on unlabeled graphs.
  Our result has an interesting consequence in the realm of digraph width
measures: Strengthening the recent result, we show that no subdigraph-monotone
measure can be "algorithmically useful", unless it is within a poly-logarithmic
factor of undirected tree-width.

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The categories of this paper are: ao-sci
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The categories of this paper are: stat
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The categories of this paper are: bayes-an
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The categories of this paper are: quant-ph
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The categories of this paper are: funct-an
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The categories of this paper are: comp-gas
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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: plasm-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: gr-qc
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The categories of this paper are: econ
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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: q-alg
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The categories of this paper are: supr-con