73 research outputs found
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; 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 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
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
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
Lower Bounds on the Complexity of MSO_1 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 (MSO2) is decidable in linear time on any class of graphs of bounded tree-width. In the parlance of parameterized complexity, this means that MSO2 model-checking is fixed-parameter tractable with respect to the tree-width as parameter. 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 a 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
Parameterized Complexity of Fair Vertex Evaluation Problems
A prototypical graph problem is centered around a graph-theoretic property for a set of vertices and a solution to it is a set of vertices for which the desired property holds. The task is to decide whether, in the given graph, there exists a solution of a certain quality, where we use size as a quality measure. In this work, we are changing the measure to the fair measure (cf. Lin and Sahni [Li-Shin Lin and Sartaj Sahni, 1989]). The fair measure of a set of vertices S is (at most) k if the number of neighbors in the set S of any vertex (in the input graph) does not exceed k. One possible way to study graph problems is by defining the property in a certain logic. For a given objective, an evaluation problem is to find a set (of vertices) that simultaneously minimizes the assumed measure and satisfies an appropriate formula. More formally, we study the {MSO} Fair Vertex Evaluation, where the graph-theoretic property is described by an {MSO} formula.
In the presented paper we show that there is an FPT algorithm for the {MSO} Fair Vertex Evaluation problem for formulas with one free variable parameterized by the twin cover number of the input graph and the size of the formula. One may define an extended variant of {MSO} Fair Vertex Evaluation for formulas with l free variables; here we measure a maximum number of neighbors in each of the l sets. However, such variant is {W[1]}-hard for parameter l even on graphs with twin cover one.
Furthermore, we study the Fair Vertex Cover (Fair VC) problem. Fair VC is among the simplest problems with respect to the demanded property (i.e., the rest forms an edgeless graph). On the negative side, Fair VC is {W[1]}-hard when parameterized by both treedepth and feedback vertex set of the input graph. On the positive side, we provide an FPT algorithm for the parameter modular width
Expanding the expressive power of Monadic Second-Order logic on restricted graph classes
We combine integer linear programming and recent advances in Monadic
Second-Order model checking to obtain two new algorithmic meta-theorems for
graphs of bounded vertex-cover. The first shows that cardMSO1, an extension of
the well-known Monadic Second-Order logic by the addition of cardinality
constraints, can be solved in FPT time parameterized by vertex cover. The
second meta-theorem shows that the MSO partitioning problems introduced by Rao
can also be solved in FPT time with the same parameter. The significance of our
contribution stems from the fact that these formalisms can describe problems
which are W[1]-hard and even NP-hard on graphs of bounded tree-width.
Additionally, our algorithms have only an elementary dependence on the
parameter and formula. We also show that both results are easily extended from
vertex cover to neighborhood diversity.Comment: Accepted for IWOCA 201
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