76 research outputs found
Model Checking Lower Bounds for Simple Graphs
A well-known result by Frick and Grohe shows that deciding FO logic on trees
involves a parameter dependence that is a tower of exponentials. Though this
lower bound is tight for Courcelle's theorem, it has been evaded by a series of
recent meta-theorems for other graph classes. Here we provide some additional
non-elementary lower bound results, which are in some senses stronger. Our goal
is to explain common traits in these recent meta-theorems and identify barriers
to further progress. More specifically, first, we show that on the class of
threshold graphs, and therefore also on any union and complement-closed class,
there is no model-checking algorithm with elementary parameter dependence even
for FO logic. Second, we show that there is no model-checking algorithm with
elementary parameter dependence for MSO logic even restricted to paths (or
equivalently to unary strings), unless E=NE. As a corollary, we resolve an open
problem on the complexity of MSO model-checking on graphs of bounded max-leaf
number. Finally, we look at MSO on the class of colored trees of depth d. We
show that, assuming the ETH, for every fixed d>=1 at least d+1 levels of
exponentiation are necessary for this problem, thus showing that the (d+1)-fold
exponential algorithm recently given by Gajarsk\`{y} and Hlin\u{e}n\`{y} is
essentially optimal
Parameterized Complexity of 1-Planarity
We consider the problem of finding a 1-planar drawing for a general graph,
where a 1-planar drawing is a drawing in which each edge participates in at
most one crossing. Since this problem is known to be NP-hard we investigate the
parameterized complexity of the problem with respect to the vertex cover
number, tree-depth, and cyclomatic number. For these parameters we construct
fixed-parameter tractable algorithms. However, the problem remains NP-complete
for graphs of bounded bandwidth, pathwidth, or treewidth.Comment: WADS 201
Twin-width I: tractable FO model checking
Inspired by a width invariant defined on permutations by Guillemot and Marx
[SODA '14], we introduce the notion of twin-width on graphs and on matrices.
Proper minor-closed classes, bounded rank-width graphs, map graphs, -free
unit -dimensional ball graphs, posets with antichains of bounded size, and
proper subclasses of dimension-2 posets all have bounded twin-width. On all
these classes (except map graphs without geometric embedding) we show how to
compute in polynomial time a sequence of -contractions, witness that the
twin-width is at most . We show that FO model checking, that is deciding if
a given first-order formula evaluates to true for a given binary
structure on a domain , is FPT in on classes of bounded
twin-width, provided the witness is given. More precisely, being given a
-contraction sequence for , our algorithm runs in time where is a computable but non-elementary function. We also prove that
bounded twin-width is preserved by FO interpretations and transductions
(allowing operations such as squaring or complementing a graph). This unifies
and significantly extends the knowledge on fixed-parameter tractability of FO
model checking on non-monotone classes, such as the FPT algorithm on
bounded-width posets by Gajarsk\'y et al. [FOCS '15].Comment: 49 pages, 9 figure
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