46 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
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
Beyond Hypertree Width: Decomposition Methods Without Decompositions
The general intractability of the constraint satisfaction problem has
motivated the study of restrictions on this problem that permit polynomial-time
solvability. One major line of work has focused on structural restrictions,
which arise from restricting the interaction among constraint scopes. In this
paper, we engage in a mathematical investigation of generalized hypertree
width, a structural measure that has up to recently eluded study. We obtain a
number of computational results, including a simple proof of the tractability
of CSP instances having bounded generalized hypertree width
Approximating acyclicity parameters of sparse hypergraphs
The notions of hypertree width and generalized hypertree width were
introduced by Gottlob, Leone, and Scarcello in order to extend the concept of
hypergraph acyclicity. These notions were further generalized by Grohe and
Marx, who introduced the fractional hypertree width of a hypergraph. All these
width parameters on hypergraphs are useful for extending tractability of many
problems in database theory and artificial intelligence. In this paper, we
study the approximability of (generalized, fractional) hyper treewidth of
sparse hypergraphs where the criterion of sparsity reflects the sparsity of
their incidence graphs. Our first step is to prove that the (generalized,
fractional) hypertree width of a hypergraph H is constant-factor sandwiched by
the treewidth of its incidence graph, when the incidence graph belongs to some
apex-minor-free graph class. This determines the combinatorial borderline above
which the notion of (generalized, fractional) hypertree width becomes
essentially more general than treewidth, justifying that way its functionality
as a hypergraph acyclicity measure. While for more general sparse families of
hypergraphs treewidth of incidence graphs and all hypertree width parameters
may differ arbitrarily, there are sparse families where a constant factor
approximation algorithm is possible. In particular, we give a constant factor
approximation polynomial time algorithm for (generalized, fractional) hypertree
width on hypergraphs whose incidence graphs belong to some H-minor-free graph
class
Panel on “Past and future of computer science theory”
The twenty-ninth edition of the SEBD (Italian Symposium on Advanced Database Systems), held on 5-9 September 2021 in Pizzo (Calabria Region, Italy), included a joint seminar on “Reminiscence of TIDB 1981” with invited talks given by some of the participants to the Advanced Seminar on Theoretical Issues in Databases (TIDB), which took place in the same region exactly forty years earlier. The joint seminar was concluded by a Panel on “The Past and the Future of Computer Science Theory” with the participation of four distinguished computer science theorists (Ronald Fagin, Georg Gottlob, Christos Papadimitriou and Moshe Vardi), who were interviewed by Giorgio Ausiello, Maurizio Lenzerini, Luigi Palopoli, Domenico Saccà and Francesco Scarcello. This paper reports the summaries of the four interviews
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
Approximating Acyclicity Parameters of Sparse Hypergraphs
The notions of hypertree width and generalized hypertree width were introduced by Gottlob, Leone, and Scarcello (PODS'99, PODS'01) in order to extend the concept of hypergraph acyclicity. These notions were further generalized by Grohe and Marx in SODA'06, who introduced the fractional hypertree width of a hypergraph. All these width parameters on hypergraphs are useful for extending tractability of many problems in database theory and artificial intelligence. Computing each of these width parameters is known to be an NP-hard problem. Moreover, the (generalized) hypertree width of an n-vertex hypergraph cannot be approximated within a logarithmic factor unless P=NP. In this paper, we study the approximability of (generalized, fractional) hyper treewidth of sparse hypergraphs where the criterion of sparsity reflects the sparsity of their incidence graphs. Our first step is to prove that the (generalized, fractional) hypertree width of a hypergraph is constant-factor sandwiched by the treewidth of its incidence graph, when the incidence graph belongs to some apex-minor-free graph class (the family of apex-minor-free graph classes includes planar graphs and graphs of bounded genus). This determines the combinatorial borderline above which the notion of (generalized, fractional) hypertree width becomes essentially more general than treewidth, justifying that way its functionality as a hypergraph acyclicity measure. While for more general sparse families of hypergraphs treewidth of incidence graphs and all hypertree width parameters may differ arbitrarily, there are sparse families where a constant factor approximation algorithm is possible. In particular, we give a constant factor approximation polynomial time algorithm for (generalized, fractional) hypertree width on hypergraphs whose incidence graphs belong to some H-minor-free graph class. This extends the results of Feige, Hajiaghayi, and Lee from STOC'05 on approximating treewidth of H-minor-free graphs.publishedVersio