1,150 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
First-order queries on classes of structures with bounded expansion
We consider the evaluation of first-order queries over classes of databases
with bounded expansion. The notion of bounded expansion is fairly broad and
generalizes bounded degree, bounded treewidth and exclusion of at least one
minor. It was known that over a class of databases with bounded expansion,
first-order sentences could be evaluated in time linear in the size of the
database. We give a different proof of this result. Moreover, we show that
answers to first-order queries can be enumerated with constant delay after a
linear time preprocessing. We also show that counting the number of answers to
a query can be done in time linear in the size of the database
Constant Delay Enumeration with FPT-Preprocessing for Conjunctive Queries of Bounded Submodular Width
Marx (STOC 2010, J. ACM 2013) introduced the notion of submodular width of a conjunctive query (CQ) and showed that for any class Phi of Boolean CQs of bounded submodular width, the model-checking problem for Phi on the class of all finite structures is fixed-parameter tractable (FPT). Note that for non-Boolean queries, the size of the query result may be far too large to be computed entirely within FPT time. We investigate the free-connex variant of submodular width and generalise Marx\u27s result to non-Boolean queries as follows: For every class Phi of CQs of bounded free-connex submodular width, within FPT-preprocessing time we can build a data structure that allows to enumerate, without repetition and with constant delay, all tuples of the query result. Our proof builds upon Marx\u27s splitting routine to decompose the query result into a union of results; but we have to tackle the additional technical difficulty to ensure that these can be enumerated efficiently
Enumeration of the Monomials of a Polynomial and Related Complexity Classes
We study the problem of generating monomials of a polynomial in the context
of enumeration complexity. In this setting, the complexity measure is the delay
between two solutions and the total time. We present two new algorithms for
restricted classes of polynomials, which have a good delay and the same global
running time as the classical ones. Moreover they are simple to describe, use
little evaluation points and one of them is parallelizable. We introduce three
new complexity classes, TotalPP, IncPP and DelayPP, which are probabilistic
counterparts of the most common classes for enumeration problems, hoping that
randomization will be a tool as strong for enumeration as it is for decision.
Our interpolation algorithms proves that a lot of interesting problems are in
these classes like the enumeration of the spanning hypertrees of a 3-uniform
hypergraph.
Finally we give a method to interpolate a degree 2 polynomials with an
acceptable (incremental) delay. We also prove that finding a specified monomial
in a degree 2 polynomial is hard unless RP = NP. It suggests that there is no
algorithm with a delay as good (polynomial) as the one we achieve for
multilinear polynomials
First-Order Query Evaluation with Cardinality Conditions
We study an extension of first-order logic that allows to express cardinality
conditions in a similar way as SQL's COUNT operator. The corresponding logic
FOC(P) was introduced by Kuske and Schweikardt (LICS'17), who showed that query
evaluation for this logic is fixed-parameter tractable on classes of structures
(or databases) of bounded degree. In the present paper, we first show that the
fixed-parameter tractability of FOC(P) cannot even be generalised to very
simple classes of structures of unbounded degree such as unranked trees or
strings with a linear order relation.
Then we identify a fragment FOC1(P) of FOC(P) which is still sufficiently
strong to express standard applications of SQL's COUNT operator. Our main
result shows that query evaluation for FOC1(P) is fixed-parameter tractable
with almost linear running time on nowhere dense classes of structures. As a
corollary, we also obtain a fixed-parameter tractable algorithm for counting
the number of tuples satisfying a query over nowhere dense classes of
structures
On The Power of Tree Projections: Structural Tractability of Enumerating CSP Solutions
The problem of deciding whether CSP instances admit solutions has been deeply
studied in the literature, and several structural tractability results have
been derived so far. However, constraint satisfaction comes in practice as a
computation problem where the focus is either on finding one solution, or on
enumerating all solutions, possibly projected to some given set of output
variables. The paper investigates the structural tractability of the problem of
enumerating (possibly projected) solutions, where tractability means here
computable with polynomial delay (WPD), since in general exponentially many
solutions may be computed. A general framework based on the notion of tree
projection of hypergraphs is considered, which generalizes all known
decomposition methods. Tractability results have been obtained both for classes
of structures where output variables are part of their specification, and for
classes of structures where computability WPD must be ensured for any possible
set of output variables. These results are shown to be tight, by exhibiting
dichotomies for classes of structures having bounded arity and where the tree
decomposition method is considered
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