90,723 research outputs found
A Galois Connection for Weighted (Relational) Clones of Infinite Size
A Galois connection between clones and relational clones on a fixed finite
domain is one of the cornerstones of the so-called algebraic approach to the
computational complexity of non-uniform Constraint Satisfaction Problems
(CSPs). Cohen et al. established a Galois connection between finitely-generated
weighted clones and finitely-generated weighted relational clones [SICOMP'13],
and asked whether this connection holds in general. We answer this question in
the affirmative for weighted (relational) clones with real weights and show
that the complexity of the corresponding valued CSPs is preserved
Necessary conditions for tractability of valued CSPs
The connection between constraint languages and clone theory has been a
fruitful line of research on the complexity of constraint satisfaction
problems. In a recent result, Cohen et al. [SICOMP'13] have characterised a
Galois connection between valued constraint languages and so-called weighted
clones. In this paper, we study the structure of weighted clones. We extend the
results of Creed and Zivny from [CP'11/SICOMP'13] on types of weightings
necessarily contained in every nontrivial weighted clone. This result has
immediate computational complexity consequences as it provides necessary
conditions for tractability of weighted clones and thus valued constraint
languages. We demonstrate that some of the necessary conditions are also
sufficient for tractability, while others are provably not.Comment: To appear in SIAM Journal on Discrete Mathematics (SIDMA
The power of Sherali-Adams relaxations for general-valued CSPs
We give a precise algebraic characterisation of the power of Sherali-Adams
relaxations for solvability of valued constraint satisfaction problems to
optimality. The condition is that of bounded width which has already been shown
to capture the power of local consistency methods for decision CSPs and the
power of semidefinite programming for robust approximation of CSPs.
Our characterisation has several algorithmic and complexity consequences. On
the algorithmic side, we show that several novel and many known valued
constraint languages are tractable via the third level of the Sherali-Adams
relaxation. For the known languages, this is a significantly simpler algorithm
than the previously obtained ones. On the complexity side, we obtain a
dichotomy theorem for valued constraint languages that can express an injective
unary function. This implies a simple proof of the dichotomy theorem for
conservative valued constraint languages established by Kolmogorov and Zivny
[JACM'13], and also a dichotomy theorem for the exact solvability of
Minimum-Solution problems. These are generalisations of Minimum-Ones problems
to arbitrary finite domains. Our result improves on several previous
classifications by Khanna et al. [SICOMP'00], Jonsson et al. [SICOMP'08], and
Uppman [ICALP'13].Comment: Full version of an ICALP'15 paper (arXiv:1502.05301
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Evaluating aggregate functions on possibilistic data
The need for extending information management systems to handle the imprecision of information found in the real world has been recognized. Fuzzy set theory together with possibility theory represent a uniform framework for extending the relational database model with these features. However, none of the existing proposals for handling imprecision in the literature has dealt with queries involving a functional evaluation of a set of items, traditionally referred to as aggregation. Two kinds of aggregate operators, namely, scalar aggregates and aggregate functions, exist. Both are important for most real-world applications, and are thus being supported by traditional languages like SQL or QUEL. This paper presents a framework for handling these two types of aggregates in the context of imprecise information. We consider three cases, specifically, aggregates within vague queries on precise data, aggregates within precisely specified queries on possibilistic data, and aggregates within vague queries on imprecise data. These extensions are based on fuzzy set-theoretical concepts such as the extension principle, the sigma-count operation, and the possibilistic expected value. The consistency and completeness of the proposed operations is shown
On tractability and congruence distributivity
Constraint languages that arise from finite algebras have recently been the
object of study, especially in connection with the Dichotomy Conjecture of
Feder and Vardi. An important class of algebras are those that generate
congruence distributive varieties and included among this class are lattices,
and more generally, those algebras that have near-unanimity term operations. An
algebra will generate a congruence distributive variety if and only if it has a
sequence of ternary term operations, called Jonsson terms, that satisfy certain
equations.
We prove that constraint languages consisting of relations that are invariant
under a short sequence of Jonsson terms are tractable by showing that such
languages have bounded relational width
Hybrid VCSPs with crisp and conservative valued templates
A constraint satisfaction problem (CSP) is a problem of computing a
homomorphism between two relational
structures. Analyzing its complexity has been a very fruitful research
direction, especially for fixed template CSPs, denoted , in
which the right side structure is fixed and the left side
structure is unconstrained.
Recently, the hybrid setting, written ,
where both sides are restricted simultaneously, attracted some attention. It
assumes that is taken from a class of relational structures
that additionally is closed under inverse homomorphisms. The last
property allows to exploit algebraic tools that have been developed for fixed
template CSPs. The key concept that connects hybrid CSPs with fixed-template
CSPs is the so called "lifted language". Namely, this is a constraint language
that can be constructed from an input . The
tractability of that language for any input is a
necessary condition for the tractability of the hybrid problem.
In the first part we investigate templates for which the
latter condition is not only necessary, but also is sufficient. We call such
templates widely tractable. For this purpose, we construct from
a new finite relational structure and define
as a class of structures homomorphic to . We
prove that wide tractability is equivalent to the tractability of
. Our proof is based on the key observation
that is homomorphic to if and only if the core of
is preserved by a Siggers polymorphism. Analogous
result is shown for valued conservative CSPs.Comment: 21 pages. arXiv admin note: text overlap with arXiv:1504.0706
Relational Symbolic Execution
Symbolic execution is a classical program analysis technique used to show
that programs satisfy or violate given specifications. In this work we
generalize symbolic execution to support program analysis for relational
specifications in the form of relational properties - these are properties
about two runs of two programs on related inputs, or about two executions of a
single program on related inputs. Relational properties are useful to formalize
notions in security and privacy, and to reason about program optimizations. We
design a relational symbolic execution engine, named RelSym which supports
interactive refutation, as well as proving of relational properties for
programs written in a language with arrays and for-like loops
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