66 research outputs found
The Complexity of Combinations of Qualitative Constraint Satisfaction Problems
The CSP of a first-order theory is the problem of deciding for a given
finite set of atomic formulas whether is satisfiable. Let
and be two theories with countably infinite models and disjoint
signatures. Nelson and Oppen presented conditions that imply decidability (or
polynomial-time decidability) of under the
assumption that and are decidable (or
polynomial-time decidable). We show that for a large class of
-categorical theories the Nelson-Oppen conditions are not
only sufficient, but also necessary for polynomial-time tractability of
(unless P=NP)
On the Descriptive Complexity of Temporal Constraint Satisfaction Problems
Finite-domain constraint satisfaction problems are either solvable by
Datalog, or not even expressible in fixed-point logic with counting. The border
between the two regimes coincides with an important dichotomy in universal
algebra; in particular, the border can be described by a strong height-one
Maltsev condition. For infinite-domain CSPs, the situation is more complicated
even if the template structure of the CSP is model-theoretically tame. We prove
that there is no Maltsev condition that characterizes Datalog already for the
CSPs of first-order reducts of (Q;<); such CSPs are called temporal CSPs and
are of fundamental importance in infinite-domain constraint satisfaction. Our
main result is a complete classification of temporal CSPs that can be expressed
in one of the following logical formalisms: Datalog, fixed-point logic (with or
without counting), or fixed-point logic with the Boolean rank operator. The
classification shows that many of the equivalent conditions in the finite fail
to capture expressibility in Datalog or fixed-point logic already for temporal
CSPs.Comment: 57 page
Tractable Combinations of Temporal CSPs
The constraint satisfaction problem (CSP) of a first-order theory T is the
computational problem of deciding whether a given conjunction of atomic
formulas is satisfiable in some model of T. We study the computational
complexity of CSP where and are theories with
disjoint finite relational signatures. We prove that if and are the
theories of temporal structures, i.e., structures where all relations have a
first-order definition in , then CSP is in P or
NP-complete. To this end we prove a purely algebraic statement about the
structure of the lattice of locally closed clones over the domain that
contain Aut
The Complexity of Combinations of Qualitative Constraint Satisfaction Problems
The CSP of a first-order theory is the problem of deciding for a given
finite set of atomic formulas whether is satisfiable. Let
and be two theories with countably infinite models and disjoint
signatures. Nelson and Oppen presented conditions that imply decidability (or
polynomial-time decidability) of under the
assumption that and are decidable (or
polynomial-time decidable). We show that for a large class of
-categorical theories the Nelson-Oppen conditions are not
only sufficient, but also necessary for polynomial-time tractability of
(unless P=NP).Comment: Version 2: stronger main result with better presentation of the
proof; multiple improvements in other proofs; new section structure; new
example
Query Answering in DL-Lite with Datatypes: A Non-Uniform Approach
Adding datatypes to ontology-mediated queries (OMQs) often makes query answering hard. As a consequence, the use of datatypes in OWL 2 QL has been severely restricted. In this paper we propose a new, non-uniform, way of analyzing the data-complexity of OMQ answering with datatypes. Instead of restricting the ontology language we aim at a classification of the patterns of datatype atoms in OMQs into those that can occur in non-tractable OMQs and those that only occur in tractable OMQs. To this end we establish a close link between OMQ answering with datatypes and constraint satisfaction problems over the datatypes. In a case study we apply this link to prove a P/coNP-dichotomy for OMQs over DL-Lite extended with the datatype (Q,<=). The proof employs a recent dichotomy result by Bodirsky and Kára for temporal constraint satisfaction problems
Parameterized Complexity Classification for Interval Constraints
Constraint satisfaction problems form a nicely behaved class of problems that
lends itself to complexity classification results. From the point of view of
parameterized complexity, a natural task is to classify the parameterized
complexity of MinCSP problems parameterized by the number of unsatisfied
constraints. In other words, we ask whether we can delete at most
constraints, where is the parameter, to get a satisfiable instance. In this
work, we take a step towards classifying the parameterized complexity for an
important infinite-domain CSP: Allen's interval algebra (IA). This CSP has
closed intervals with rational endpoints as domain values and employs a set
of 13 basic comparison relations such as ``precedes'' or ``during'' for
relating intervals. IA is a highly influential and well-studied formalism
within AI and qualitative reasoning that has numerous applications in, for
instance, planning, natural language processing and molecular biology. We
provide an FPT vs. W[1]-hard dichotomy for MinCSP for all . IA is sometimes extended with unions of the relations in or
first-order definable relations over , but extending our results to these
cases would require first solving the parameterized complexity of Directed
Symmetric Multicut, which is a notorious open problem. Already in this limited
setting, we uncover connections to new variants of graph cut and separation
problems. This includes hardness proofs for simultaneous cuts or feedback arc
set problems in directed graphs, as well as new tractable cases with algorithms
based on the recently introduced flow augmentation technique. Given the
intractability of MinCSP in general, we then consider (parameterized)
approximation algorithms and present a factor- fpt-approximation algorithm
Parameterized Complexity of Equality MinCSP
We study the parameterized complexity of MinCSP for so-called equality languages, i.e., for finite languages over an infinite domain such as ?, where the relations are defined via first-order formulas whose only predicate is =. This is an important class of languages that forms the starting point of all study of infinite-domain CSPs under the commonly used approach pioneered by Bodirsky, i.e., languages defined as reducts of finitely bounded homogeneous structures. Moreover, MinCSP over equality languages forms a natural class of optimisation problems in its own right, covering such problems as Edge Multicut, Steiner Multicut and (under singleton expansion) Edge Multiway Cut. We classify MinCSP(?) for every finite equality language ?, under the natural parameter, as either FPT, W[1]-hard but admitting a constant-factor FPT-approximation, or not admitting a constant-factor FPT-approximation unless FPT=W[2]. In particular, we describe an FPT case that slightly generalises Multicut, and show a constant-factor FPT-approximation for Disjunctive Multicut, the generalisation of Multicut where the "cut requests" come as disjunctions over O(1) individual cut requests s_i ? t_i. We also consider singleton expansions of equality languages, enriching an equality language with the capability for assignment constraints (x = i) for either a finite or infinitely many constants i, and fully characterize the complexity of the resulting MinCSP
The complexity of disjunctive linear Diophantine constraints.
We study the Constraint Satisfaction Problem CSP( A), where A is first-order definable in (Z;+,1) and contains +. We prove such problems are either in P or NP-complete
Homogeneity and Homogenizability: Hard Problems for the Logic SNP
We show that the question whether a given SNP sentence defines a
homogenizable class of finite structures is undecidable, even if the sentence
comes from the connected Datalog fragment and uses at most binary relation
symbols. As a byproduct of our proof, we also get the undecidability of some
other properties for Datalog programs, e.g., whether they can be rewritten in
MMSNP, whether they solve some finite-domain CSP, or whether they define the
age of a reduct of a homogeneous Ramsey structure in a finite relational
signature. We subsequently show that the closely related problem of testing the
amalgamation property for finitely bounded classes is EXPSPACE-hard or
PSPACE-hard, depending on whether the input is specified by a universal
sentence or a set of forbidden substructures.Comment: 34 pages, 3 figure
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