703 research outputs found
On the Monadic Second-Order Transduction Hierarchy
We compare classes of finite relational structures via monadic second-order
transductions. More precisely, we study the preorder where we set C \subseteq K
if, and only if, there exists a transduction {\tau} such that
C\subseteq{\tau}(K). If we only consider classes of incidence structures we can
completely describe the resulting hierarchy. It is linear of order type
{\omega}+3. Each level can be characterised in terms of a suitable variant of
tree-width. Canonical representatives of the various levels are: the class of
all trees of height n, for each n \in N, of all paths, of all trees, and of all
grids
Regular Combinators for String Transformations
We focus on (partial) functions that map input strings to a monoid such as
the set of integers with addition and the set of output strings with
concatenation. The notion of regularity for such functions has been defined
using two-way finite-state transducers, (one-way) cost register automata, and
MSO-definable graph transformations. In this paper, we give an algebraic and
machine-independent characterization of this class analogous to the definition
of regular languages by regular expressions. When the monoid is commutative, we
prove that every regular function can be constructed from constant functions
using the combinators of choice, split sum, and iterated sum, that are analogs
of union, concatenation, and Kleene-*, respectively, but enforce unique (or
unambiguous) parsing. Our main result is for the general case of
non-commutative monoids, which is of particular interest for capturing regular
string-to-string transformations for document processing. We prove that the
following additional combinators suffice for constructing all regular
functions: (1) the left-additive versions of split sum and iterated sum, which
allow transformations such as string reversal; (2) sum of functions, which
allows transformations such as copying of strings; and (3) function
composition, or alternatively, a new concept of chained sum, which allows
output values from adjacent blocks to mix.Comment: This is the full version, with omitted proofs and constructions, of
the conference paper currently in submissio
Are there any good digraph width measures?
Several different measures for digraph width have appeared in the last few
years. However, none of them shares all the "nice" properties of treewidth:
First, being \emph{algorithmically useful} i.e. admitting polynomial-time
algorithms for all \MS1-definable problems on digraphs of bounded width. And,
second, having nice \emph{structural properties} i.e. being monotone under
taking subdigraphs and some form of arc contractions. As for the former,
(undirected) \MS1 seems to be the least common denominator of all reasonably
expressive logical languages on digraphs that can speak about the edge/arc
relation on the vertex set.The latter property is a necessary condition for a
width measure to be characterizable by some version of the cops-and-robber game
characterizing the ordinary treewidth. Our main result is that \emph{any
reasonable} algorithmically useful and structurally nice digraph measure cannot
be substantially different from the treewidth of the underlying undirected
graph. Moreover, we introduce \emph{directed topological minors} and argue that
they are the weakest useful notion of minors for digraphs
Deciding first-order properties of nowhere dense graphs
Nowhere dense graph classes, introduced by Nesetril and Ossona de Mendez,
form a large variety of classes of "sparse graphs" including the class of
planar graphs, actually all classes with excluded minors, and also bounded
degree graphs and graph classes of bounded expansion.
We show that deciding properties of graphs definable in first-order logic is
fixed-parameter tractable on nowhere dense graph classes. At least for graph
classes closed under taking subgraphs, this result is optimal: it was known
before that for all classes C of graphs closed under taking subgraphs, if
deciding first-order properties of graphs in C is fixed-parameter tractable,
then C must be nowhere dense (under a reasonable complexity theoretic
assumption).
As a by-product, we give an algorithmic construction of sparse neighbourhood
covers for nowhere dense graphs. This extends and improves previous
constructions of neighbourhood covers for graph classes with excluded minors.
At the same time, our construction is considerably simpler than those. Our
proofs are based on a new game-theoretic characterisation of nowhere dense
graphs that allows for a recursive version of locality-based algorithms on
these classes. On the logical side, we prove a "rank-preserving" version of
Gaifman's locality theorem.Comment: 30 page
Between Treewidth and Clique-width
Many hard graph problems can be solved efficiently when restricted to graphs
of bounded treewidth, and more generally to graphs of bounded clique-width. But
there is a price to be paid for this generality, exemplified by the four
problems MaxCut, Graph Coloring, Hamiltonian Cycle and Edge Dominating Set that
are all FPT parameterized by treewidth but none of which can be FPT
parameterized by clique-width unless FPT = W[1], as shown by Fomin et al [7,
8]. We therefore seek a structural graph parameter that shares some of the
generality of clique-width without paying this price. Based on splits, branch
decompositions and the work of Vatshelle [18] on Maximum Matching-width, we
consider the graph parameter sm-width which lies between treewidth and
clique-width. Some graph classes of unbounded treewidth, like
distance-hereditary graphs, have bounded sm-width. We show that MaxCut, Graph
Coloring, Hamiltonian Cycle and Edge Dominating Set are all FPT parameterized
by sm-width
A Natural Generalization of Bounded Tree-Width and Bounded Clique-Width
We investigate a new width parameter, the fusion-width of a graph. It is a
natural generalization of the tree-width, yet strong enough that not only
graphs of bounded tree-width, but also graphs of bounded clique-width,
trivially have bounded fusion-width. In particular, there is no exponential
growth between tree-width and fusion-width, as is the case between tree-width
and clique-width. The new parameter gives a good intuition about the
relationship between tree-width and clique-width.Comment: To appear in the proceedings of Latin 2014. Springer LNCS 839
Proofs of partial correctness for attribute grammars with applications to recursive procedures and logic programming
AbstractAn extension of the inductive assertion method allowing one to prove the partial correctness of an attribute grammar w.r.t. a specification is presented. It is complete in an abstract sense. It is also shown that the semantics of systems of recursive imperative procedures or of recursive applicative procedures computed with call-by-value or call-by-name can be expressed by an attribute grammar associating attributes with the nodes of the so-called trees of calls. Hence the proof methods for the partial correctness of attribute grammars can be applied to these recursive procedures. We show also how the proof method can be applied in logic programming
Compact Labelings For Efficient First-Order Model-Checking
We consider graph properties that can be checked from labels, i.e., bit
sequences, of logarithmic length attached to vertices. We prove that there
exists such a labeling for checking a first-order formula with free set
variables in the graphs of every class that is \emph{nicely locally
cwd-decomposable}. This notion generalizes that of a \emph{nicely locally
tree-decomposable} class. The graphs of such classes can be covered by graphs
of bounded \emph{clique-width} with limited overlaps. We also consider such
labelings for \emph{bounded} first-order formulas on graph classes of
\emph{bounded expansion}. Some of these results are extended to counting
queries
Parameterized Approximation Schemes using Graph Widths
Combining the techniques of approximation algorithms and parameterized
complexity has long been considered a promising research area, but relatively
few results are currently known. In this paper we study the parameterized
approximability of a number of problems which are known to be hard to solve
exactly when parameterized by treewidth or clique-width. Our main contribution
is to present a natural randomized rounding technique that extends well-known
ideas and can be used for both of these widths. Applying this very generic
technique we obtain approximation schemes for a number of problems, evading
both polynomial-time inapproximability and parameterized intractability bounds
A SAT Approach to Clique-Width
Clique-width is a graph invariant that has been widely studied in
combinatorics and computer science. However, computing the clique-width of a
graph is an intricate problem, the exact clique-width is not known even for
very small graphs. We present a new method for computing the clique-width of
graphs based on an encoding to propositional satisfiability (SAT) which is then
evaluated by a SAT solver. Our encoding is based on a reformulation of
clique-width in terms of partitions that utilizes an efficient encoding of
cardinality constraints. Our SAT-based method is the first to discover the
exact clique-width of various small graphs, including famous graphs from the
literature as well as random graphs of various density. With our method we
determined the smallest graphs that require a small pre-described clique-width.Comment: proofs in section 3 updated, results remain unchange
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