6,674 research outputs found
Shrub-depth: Capturing Height of Dense Graphs
The recent increase of interest in the graph invariant called tree-depth and
in its applications in algorithms and logic on graphs led to a natural
question: is there an analogously useful "depth" notion also for dense graphs
(say; one which is stable under graph complementation)? To this end, in a 2012
conference paper, a new notion of shrub-depth has been introduced, such that it
is related to the established notion of clique-width in a similar way as
tree-depth is related to tree-width. Since then shrub-depth has been
successfully used in several research papers. Here we provide an in-depth
review of the definition and basic properties of shrub-depth, and we focus on
its logical aspects which turned out to be most useful. In particular, we use
shrub-depth to give a characterization of the lower levels of the
MSO1 transduction hierarchy of simple graphs
Recovering sparse graphs
We construct a fixed parameter algorithm parameterized by d and k that takes
as an input a graph G' obtained from a d-degenerate graph G by complementing on
at most k arbitrary subsets of the vertex set of G and outputs a graph H such
that G and H agree on all but f(d,k) vertices.
Our work is motivated by the first order model checking in graph classes that
are first order interpretable in classes of sparse graphs. We derive as a
corollary that if G_0 is a graph class with bounded expansion, then the first
order model checking is fixed parameter tractable in the class of all graphs
that can obtained from a graph G from G_0 by complementing on at most k
arbitrary subsets of the vertex set of G; this implies an earlier result that
the first order model checking is fixed parameter tractable in graph classes
interpretable in classes of graphs with bounded maximum degree
Quantified Conjunctive Queries on Partially Ordered Sets
We study the computational problem of checking whether a quantified
conjunctive query (a first-order sentence built using only conjunction as
Boolean connective) is true in a finite poset (a reflexive, antisymmetric, and
transitive directed graph). We prove that the problem is already NP-hard on a
certain fixed poset, and investigate structural properties of posets yielding
fixed-parameter tractability when the problem is parameterized by the query.
Our main algorithmic result is that model checking quantified conjunctive
queries on posets of bounded width is fixed-parameter tractable (the width of a
poset is the maximum size of a subset of pairwise incomparable elements). We
complement our algorithmic result by complexity results with respect to classes
of finite posets in a hierarchy of natural poset invariants, establishing its
tightness in this sense.Comment: Accepted at IPEC 201
First-Order Model-Checking in Random Graphs and Complex Networks
Complex networks are everywhere. They appear for example in the form of
biological networks, social networks, or computer networks and have been
studied extensively. Efficient algorithms to solve problems on complex networks
play a central role in today's society. Algorithmic meta-theorems show that
many problems can be solved efficiently. Since logic is a powerful tool to
model problems, it has been used to obtain very general meta-theorems. In this
work, we consider all problems definable in first-order logic and analyze which
properties of complex networks allow them to be solved efficiently.
The mathematical tool to describe complex networks are random graph models.
We define a property of random graph models called
-power-law-boundedness. Roughly speaking, a random graph is
-power-law-bounded if it does not admit strong clustering and its
degree sequence is bounded by a power-law distribution with exponent at least
(i.e. the fraction of vertices with degree is roughly
).
We solve the first-order model-checking problem (parameterized by the length
of the formula) in almost linear FPT time on random graph models satisfying
this property with . This means in particular that one can solve
every problem expressible in first-order logic in almost linear expected time
on these random graph models. This includes for example preferential attachment
graphs, Chung-Lu graphs, configuration graphs, and sparse Erd\H{o}s-R\'{e}nyi
graphs. Our results match known hardness results and generalize previous
tractability results on this topic
Model-checking for successor-invariant first-order formulas on graph classes of bounded expansion
A successor-invariant first-order formula is a formula that has access to an auxiliary successor relation on a structure's universe, but the model relation is independent of the particular interpretation of this relation. It is well known that successor-invariant formulas are more expressive on finite structures than plain first-order formulas without a successor relation. This naturally raises the question whether this increase in expressive power comes at an extra cost to solve the model-checking problem, that is, the problem to decide whether a given structure together with some (and hence every) successor relation is a model of a given formula. It was shown earlier that adding successor-invariance to first-order logic essentially comes at no extra cost for the model-checking problem on classes of finite structures whose underlying Gaifman graph is planar [1], excludes a fixed minor [2] or a fixed topological minor [3], [4]. In this work we show that the model-checking problem for successor-invariant formulas is fixed-parameter tractable on any class of finite structures whose underlying Gaifman graphs form a class of bounded expansion. Our result generalises all earlier results and comes close to the best tractability results on nowhere dense classes of graphs currently known for plain first-order logic
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