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42
Classifying the complexity of constraints using finite algebras
 SIAM Journal on Computing
, 2005
"... Abstract. Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NPcomplete in general, but certain restrictions on the form of the constraints can ensure tractability. Here we show that any set of relations used to specify th ..."
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Cited by 185 (32 self)
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Abstract. Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NPcomplete in general, but certain restrictions on the form of the constraints can ensure tractability. Here we show that any set of relations used to specify the allowed forms of constraints can be associated with a finite universal algebra and we explore how the computational complexity of the corresponding constraint satisfaction problem is connected to the properties of this algebra. Hence, we completely translate the problem of classifying the complexity of restricted constraint satisfaction problems into the language of universal algebra. We introduce a notion of “tractable algebra, ” and investigate how the tractability of an algebra relates to the tractability of the smaller algebras which may be derived from it, including its subalgebras and homomorphic images. This allows us to reduce significantly the types of algebras which need to be classified. Using our results we also show that if the decision problem associated with a given collection of constraint types can be solved efficiently, then so can the corresponding search problem. We then classify all finite strictly simple surjective algebras with respect to tractability, obtaining a dichotomy theorem which generalizes Schaefer’s dichotomy for the generalized satisfiability problem. Finally, we suggest a possible general algebraic criterion for distinguishing the tractable and intractable cases of the constraint satisfaction problem.
Datalog and constraint satisfaction with infinite templates
 In Proceedings of the 23rd International Symposium on Theoretical Aspects of Computer Science (STACS’06), LNCS 3884
, 2006
"... Abstract. On finite structures, there is a wellknown connection between the expressive power of Datalog, finite variable logics, the existential pebble game, and bounded hypertree duality. We study this connection for infinite structures. This has applications for constraint satisfaction with infin ..."
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Cited by 38 (20 self)
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Abstract. On finite structures, there is a wellknown connection between the expressive power of Datalog, finite variable logics, the existential pebble game, and bounded hypertree duality. We study this connection for infinite structures. This has applications for constraint satisfaction with infinite templates, i.e., for all computational problems that are closed under disjoint unions and whose complement is closed under homomorphisms. If the template Γ is ωcategorical, we obtain alternative characterizations of bounded Datalog width. We also show that CSP(Γ) can be solved in polynomial time if Γ is ωcategorical and the input is restricted to instances of bounded treewidth. Finally, we prove algebraic characterisations of those ωcategorical templates whose CSP has Datalog width (1, k), and for those whose CSP has strict Datalog width k.
The complexity of temporal constraint satisfaction problems
 J. ACM
"... A temporal constraint language is a set of relations that has a firstorder definition in (Q; <), the dense linear order of the rational numbers. We present a complete complexity classification of the constraint satisfaction problem (CSP) for temporal constraint languages: if the constraint langu ..."
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Cited by 34 (23 self)
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A temporal constraint language is a set of relations that has a firstorder definition in (Q; <), the dense linear order of the rational numbers. We present a complete complexity classification of the constraint satisfaction problem (CSP) for temporal constraint languages: if the constraint language is contained in one out of nine temporal constraint languages, then the CSP can be solved in polynomial time; otherwise, the CSP is NPcomplete. Our proof combines modeltheoretic concepts with techniques from universal algebra, and also applies the socalled product Ramsey theorem, which we believe will useful in similar contexts of
A survey of homogeneous structures
, 2010
"... A relational first order structure is homogeneous if it is countable (possibly finite) and every isomorphism between finite substructures extends to an automorphism. This article is a survey of several aspects of homogeneity, with emphasis on countably infinite homogeneous structures. These arise as ..."
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Cited by 28 (0 self)
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A relational first order structure is homogeneous if it is countable (possibly finite) and every isomorphism between finite substructures extends to an automorphism. This article is a survey of several aspects of homogeneity, with emphasis on countably infinite homogeneous structures. These arise as Fraissé limits of amalgamation classes of finite structures. The subject has connections to model theory, to permutation group theory, to combinatorics (for example through combinatorial enumeration, and through Ramsey theory), to descriptive set theory. Recently there has been a focus on connections to topological dynamics, and to constraint satisfaction. The article discusses connections between these topics, with an emphasis on examples, and on how special properties of an amalgamation class yield consequences for the automorphism group.
Constraint Satisfaction Problems with Countable Homogeneous Templates
"... Allowing templates with infinite domains greatly expands the range of problems that can be formulated as a nonuniform constraint satisfaction problem. It turns out that many CSPs over infinite templates can be formulated with templates that are ωcategorical. We survey examples of such problems in ..."
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Cited by 25 (10 self)
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Allowing templates with infinite domains greatly expands the range of problems that can be formulated as a nonuniform constraint satisfaction problem. It turns out that many CSPs over infinite templates can be formulated with templates that are ωcategorical. We survey examples of such problems in temporal and spatial reasoning, infinitedimensional algebra, acyclic colorings in graph theory, artificial intelligence, phylogenetic reconstruction in computational biology, and tree descriptions in computational linguistics. We then give an introduction to the universalalgebraic approach to infinitedomain constraint satisfaction, and discuss how cores, polymorphism clones, and pseudovarieties can be used to study the computational complexity of CSPs with ωcategorical templates. The theoretical results will be illustrated by examples from the mentioned application areas. We close with a series of open problems and promising directions of future research.
The core of a countably categorical structure
 In Proceedings of the 22nd Annual Symposium on Theoretical Aspects of Computer Science (STACS’05), LNCS 3404
, 2005
"... Abstract. A relational structure is a core, if all its endomorphisms are embeddings. This notion is important for computational complexity classification of constraint satisfaction problems. It is a fundamental fact that every finite structure S has a core, i.e., S has an endomorphism e such that th ..."
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Cited by 25 (19 self)
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Abstract. A relational structure is a core, if all its endomorphisms are embeddings. This notion is important for computational complexity classification of constraint satisfaction problems. It is a fundamental fact that every finite structure S has a core, i.e., S has an endomorphism e such that the structure induced by e(S) is a core; moreover, the core is unique up to isomorphism. We prove that every ωcategorical structure has a core. Moreover, every ωcategorical structure is homomorphically equivalent to a modelcomplete core, which is unique up to isomorphism, and which is finite or ωcategorical. We discuss consequences for constraint satisfaction with ωcategorical templates. 1.
The reducts of equality up to primitive positive interdefinability
 Journal of Symbolic Logic
"... Abstract. We initiate the study of reducts of relational structures up to primitive positive interdefinability: After providing the tools for such a study, we apply these tools in order to obtain a classification of the reducts of the logic of equality. It turns out that there exists a continuum of ..."
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Cited by 21 (18 self)
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Abstract. We initiate the study of reducts of relational structures up to primitive positive interdefinability: After providing the tools for such a study, we apply these tools in order to obtain a classification of the reducts of the logic of equality. It turns out that there exists a continuum of such reducts. Equivalently, expressed in the language of universal algebra, we classify those locally closed clones over a countable domain which contain all permutations of the domain. Contents
The complexity of equality constraint languages
 Cornell University
, 2006
"... Abstract. We apply the algebraic approach to infinitevalued constraint satisfaction to classify the computational complexity of all constraint satisfaction problems with templates that have a highly transitive automorphism group. A relational structure has such an automorphism group if and only if ..."
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Cited by 17 (13 self)
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Abstract. We apply the algebraic approach to infinitevalued constraint satisfaction to classify the computational complexity of all constraint satisfaction problems with templates that have a highly transitive automorphism group. A relational structure has such an automorphism group if and only if all the constraint types are Boolean combinations of the equality relation, and we call the corresponding constraint languages equality constraint languages. We show that an equality constraint language is tractable if it admits a constant unary or an injective binary polymorphism, and is NPcomplete otherwise.
ON THE SCOPE OF THE UNIVERSALALGEBRAIC APPROACH TO CONSTRAINT SATISFACTION
, 909
"... Abstract. The universalalgebraic approach has proved a powerful tool in the study of the computational complexity of constraint satisfaction problems (CSPs). This approach has previously been applied to the study of CSPs with finite or (infinite) ωcategorical templates. Our first result is an exac ..."
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Cited by 12 (9 self)
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Abstract. The universalalgebraic approach has proved a powerful tool in the study of the computational complexity of constraint satisfaction problems (CSPs). This approach has previously been applied to the study of CSPs with finite or (infinite) ωcategorical templates. Our first result is an exact characterization of those CSPs that can be formulated with (a finite or) an ωcategorical template. The universalalgebraic approach relies on the fact that in finite or ωcategorical structures A, a relation is primitive positive definable if and only if it is preserved by the polymorphisms of A. In this paper, we present results that can be used to study the computational complexity of CSPs with arbitrary infinite templates. Specifically, we prove that every CSP can be formulated with a template A such that a relation is primitive positive definable in A if and only if it is firstorder definable on A and preserved by the infinitary polymorphisms of A. We present applications of our general results to the description and analysis of the computational complexity of CSPs. In particular, we present a polymorphismbased description of those CSPs that are firstorder definable (and therefore can be solved in polynomialtime), and give general hardness criteria based on the absence of polymorphisms that depend on more than one argument. 1.
A NEW LINE OF ATTACK ON THE DICHOTOMY CONJECTURE
"... The well known dichotomy conjecture of Feder and Vardi states that for every family Γ of constraints CSP(Γ) is either polynomially solvable or NPhard. Bulatov and Jeavons reformulated this conjecture in terms of the properties of the algebra P ol(Γ), where the latter is the collection of those mar ..."
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Cited by 10 (0 self)
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The well known dichotomy conjecture of Feder and Vardi states that for every family Γ of constraints CSP(Γ) is either polynomially solvable or NPhard. Bulatov and Jeavons reformulated this conjecture in terms of the properties of the algebra P ol(Γ), where the latter is the collection of those mary operations (m = 1, 2,...) that keep all constraints in Γ invariant. We show that the algebraic condition boils down to whether there are arbitrarily resilient functions in P ol(Γ). Equivalently, we can express this in the terms of the PCP theory: CSP(Γ) is NPhard iff all long code tests created from Γ that passes with zero error admits only juntas1. Then, using this characterization and a result of Dinur, Friedgut and Regev, we give an entirely new and transparent proof to the HellNeˇsetˇril theorem, which states that for a simple connected undirected graph H, the problem CSP(H) is NPhard if and only if H is nonbipartite. We also introduce another notion of resilience (we call it strong resilience), and we use it in the investigation of CSP problems that ’do not have the ability to count. ’ The complexity of this class is unknown. Several authors conjectured that CSP problems without the ability to count have bounded width, or equivalently, that they can be characterized by existential kpebble games. The resolution of this conjecture would be a major step towards the resolution of the dichotomy conjecture. We show that CSP problems without the ability to count are exactly the ones with strongly resilient term operations, which might give a handier tool to attack the conjecture than the known algebraic characterizations.