58 research outputs found
On Low Treewidth Approximations of Conjunctive Queries
We recently initiated the study of approximations of conjunctive queries within classes that admit tractable query evaluation (with respect to combined complexity). Those include classes of acyclic, bounded treewidth, or bounded hypertreewidth queries. Such approximations are always guaranteed to exist. However, while for acyclic and bounded hypertreewidth queries we have shown a number of examples of interesting approximations, for queries of bounded treewidth the study had been restricted to queries over graphs, where such approximations usually trivialize. In this note we show that for relations of arity greater than two, the notion of low treewidth approximations is a rich one, as many queries possess them. In fact we look at approximations of queries of maximum possible treewidth by queries of minimum possible treewidth (i.e., one), and show that even in this case the structure of approximations remain rather rich as long as input relations are not binary
First-order query evaluation on structures of bounded degree
We consider the enumeration problem of first-order queries over structures of
bounded degree. It was shown that this problem is in the Constant-Delaylin
class. An enumeration problem belongs to Constant-Delaylin if for an input of
size n it can be solved by: - an O(n) precomputation phase building an index
structure, - followed by a phase enumerating the answers with no repetition and
a constant delay between two consecutive outputs. In this article we give a
different proof of this result based on Gaifman's locality theorem for
first-order logic. Moreover, the constants we obtain yield a total evaluation
time that is triply exponential in the size of the input formula, matching the
complexity of the best known evaluation algorithms
On the Complexity of Existential Positive Queries
We systematically investigate the complexity of model checking the
existential positive fragment of first-order logic. In particular, for a set of
existential positive sentences, we consider model checking where the sentence
is restricted to fall into the set; a natural question is then to classify
which sentence sets are tractable and which are intractable. With respect to
fixed-parameter tractability, we give a general theorem that reduces this
classification question to the corresponding question for primitive positive
logic, for a variety of representations of structures. This general theorem
allows us to deduce that an existential positive sentence set having bounded
arity is fixed-parameter tractable if and only if each sentence is equivalent
to one in bounded-variable logic. We then use the lens of classical complexity
to study these fixed-parameter tractable sentence sets. We show that such a set
can be NP-complete, and consider the length needed by a translation from
sentences in such a set to bounded-variable logic; we prove superpolynomial
lower bounds on this length using the theory of compilability, obtaining an
interesting type of formula size lower bound. Overall, the tools, concepts, and
results of this article set the stage for the future consideration of the
complexity of model checking on more expressive logics
The complexity of acyclic conjunctive queries revisited
In this paper, we consider first-order logic over unary functions and study
the complexity of the evaluation problem for conjunctive queries described by
such kind of formulas. A natural notion of query acyclicity for this language
is introduced and we study the complexity of a large number of variants or
generalizations of acyclic query problems in that context (Boolean or not
Boolean, with or without inequalities, comparisons, etc...). Our main results
show that all those problems are \textit{fixed-parameter linear} i.e. they can
be evaluated in time where is the
size of the query , the database size, is
the size of the output and is some function whose value depends on the
specific variant of the query problem (in some cases, is the identity
function). Our results have two kinds of consequences. First, they can be
easily translated in the relational (i.e., classical) setting. Previously known
bounds for some query problems are improved and new tractable cases are then
exhibited. Among others, as an immediate corollary, we improve a result of
\~\cite{PapadimitriouY-99} by showing that any (relational) acyclic conjunctive
query with inequalities can be evaluated in time
. A second consequence of our method is
that it provides a very natural descriptive approach to the complexity of
well-known algorithmic problems. A number of examples (such as acyclic subgraph
problems, multidimensional matching, etc...) are considered for which new
insights of their complexity are given.Comment: 30 page
Provenance Circuits for Trees and Treelike Instances (Extended Version)
Query evaluation in monadic second-order logic (MSO) is tractable on trees
and treelike instances, even though it is hard for arbitrary instances. This
tractability result has been extended to several tasks related to query
evaluation, such as counting query results [3] or performing query evaluation
on probabilistic trees [10]. These are two examples of the more general problem
of computing augmented query output, that is referred to as provenance. This
article presents a provenance framework for trees and treelike instances, by
describing a linear-time construction of a circuit provenance representation
for MSO queries. We show how this provenance can be connected to the usual
definitions of semiring provenance on relational instances [20], even though we
compute it in an unusual way, using tree automata; we do so via intrinsic
definitions of provenance for general semirings, independent of the operational
details of query evaluation. We show applications of this provenance to capture
existing counting and probabilistic results on trees and treelike instances,
and give novel consequences for probability evaluation.Comment: 48 pages. Presented at ICALP'1
Optimizing Tree Decompositions in MSO
The classic algorithm of Bodlaender and Kloks solves the following problem in linear fixed-parameter time: given a tree decomposition of a graph of (possibly suboptimal) width k, compute an optimum-width tree decomposition of the graph. In this work, we prove that this problem can also be solved in MSO in the following sense: for every positive integer k, there is an MSO transduction from tree decompositions of width k to tree decompositions of optimum width. Together with our recent results, this implies that for every k there exists an MSO transduction which inputs a graph of treewidth k, and nondeterministically outputs its tree decomposition of optimum width
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