595 research outputs found
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
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
Tractable Optimization Problems through Hypergraph-Based Structural Restrictions
Several variants of the Constraint Satisfaction Problem have been proposed
and investigated in the literature for modelling those scenarios where
solutions are associated with some given costs. Within these frameworks
computing an optimal solution is an NP-hard problem in general; yet, when
restricted over classes of instances whose constraint interactions can be
modelled via (nearly-)acyclic graphs, this problem is known to be solvable in
polynomial time. In this paper, larger classes of tractable instances are
singled out, by discussing solution approaches based on exploiting hypergraph
acyclicity and, more generally, structural decomposition methods, such as
(hyper)tree decompositions
Prediction-hardness of acyclic conjunctive queries
AbstractA conjunctive query problem is a problem to determine whether or not a tuple belongs to the answer of a conjunctive query over a database. In this paper, a tuple, a conjunctive query and a database in relational database theory are regarded as a ground atom, a nonrecursive function-free definite clause and a finite set of ground atoms, respectively, in inductive logic programming terminology. An acyclic conjunctive query problem is a conjunctive query problem with acyclicity. Concerned with the acyclic conjunctive query problem, in this paper, we present the hardness results of predicting acyclic conjunctive queries from an instance with a j-database of which predicate symbol is at most j-ary. Also we deal with two kinds of instances, a simple instance as a set of ground atoms and an extended instance as a set of pairs of a ground atom and a description. We mainly show that, from both a simple and an extended instances, acyclic conjunctive queries are not polynomial-time predictable with j-databases (j⩾3) under the cryptographic assumptions, and predicting acyclic conjunctive queries with 2-databases is as hard as predicting DNF formulas. Hence, the acyclic conjunctive queries become a natural example that the equivalence between subsumption-efficiency and efficient pac-learnability from both a simple and an extended instances collapses
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