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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.
Algebraic structures in combinatorial problems
 TECHNICAL REPORT, TECHNISCHE UNIVERSITAT DRESDEN
, 2001
"... ..."
The Complexity Of Maximal Constraint Languages
, 2001
"... Many combinatorial search problems can be expressed as "constraint satisfaction problems" using an appropriate "constraint language", that is, a set of relations over some fixed finite set of values. It is wellknown that there is a tradeoff between the expressive power of a con ..."
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Cited by 38 (9 self)
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Many combinatorial search problems can be expressed as "constraint satisfaction problems" using an appropriate "constraint language", that is, a set of relations over some fixed finite set of values. It is wellknown that there is a tradeoff between the expressive power of a constraint language and the complexity of the problems it can express. In the present paper we systematically study the complexity of all maximal constraint languages, that is, languages whose expressive power is just weaker than that of the language of all constraints. Using the algebraic invariance properties of constraints, we exhibit a strong necessary condition for tractability of such a constraint language. Moreover, we show that, at least for small sets of values, this condition is also sufficient.
The computational complexity of quantified constraint satisfaction
, 2004
"... The constraint satisfaction problem (CSP) is a framework for modelling search problems. An instance of the CSP consists of a set of variables and a set of constraints on the variables; the question is to decide whether or not there is an assignment to the variables satisfying all of the constraints. ..."
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Cited by 35 (10 self)
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The constraint satisfaction problem (CSP) is a framework for modelling search problems. An instance of the CSP consists of a set of variables and a set of constraints on the variables; the question is to decide whether or not there is an assignment to the variables satisfying all of the constraints. The quantified constraint satisfaction problem (QCSP) is a generalization of the CSP in which variables may be both universally and existentially quantified. The general intractability of the CSP and QCSP motivates the search for restricted cases of these problems that are polynomialtime tractable. In this
An Algebraic Approach To MultiSorted Constraints
 Proceedings of 9th International Conference on Principles and Practice of Constraint Programming
, 2003
"... We describe a common framework for the Constraint Satisfaction Problem and the Conjunctive Query Evaluation Problem, encompassing a generalised form of these problems in which different variables may take values from different sets. The framework we develop allows us to specify natural subclasses of ..."
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Cited by 23 (7 self)
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We describe a common framework for the Constraint Satisfaction Problem and the Conjunctive Query Evaluation Problem, encompassing a generalised form of these problems in which different variables may take values from different sets. The framework we develop allows us to specify natural subclasses of these two problems using algebraic techniques, and to establish when these subclasses are tractable. We show that a range of tractable classes can be obtained by combining recently identified tractable subclasses of the usual constraint satisfaction problem over a single set of values. We also systematically develop an algebraic structural theory for the general problem, which provides the prerequisites for further use of the powerful algebraic machinery.
Complexity of conservative Constraint Satisfaction Problems
"... In a constraint satisfaction problem (CSP) the aim is to find an assignment of values to a given set of variables, subject to specified constraints. The CSP is known to be NPcomplete in general. However, certain restrictions on the form of the allowed constraints can lead to problems solvable in po ..."
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Cited by 21 (3 self)
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In a constraint satisfaction problem (CSP) the aim is to find an assignment of values to a given set of variables, subject to specified constraints. The CSP is known to be NPcomplete in general. However, certain restrictions on the form of the allowed constraints can lead to problems solvable in polynomial time. Such restrictions are usually imposed by specifying a constraint language, that is, a set of relations that are allowed to be used as constraints. A principal research direction aims to distinguish those constraint languages that give rise to tractable CSPs from those that do not. We achieve
Recent results on the algebraic approach to the CSP
 In The Same Volume
, 2008
"... Abstract. We describe an algebraic approach to the constraint satisfaction problem (CSP) and present recent results on the CSP that make use of, in an essential way, this algebraic framework. 1 ..."
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Cited by 21 (4 self)
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Abstract. We describe an algebraic approach to the constraint satisfaction problem (CSP) and present recent results on the CSP that make use of, in an essential way, this algebraic framework. 1
Closures and dichotomies for quantified constraints
, 2004
"... 1 Introduction Constraint satisfaction problems are ubiquitous in several different areas of artificialintelligence and computer science, because constraints are widely used to specify ..."
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Cited by 5 (1 self)
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1 Introduction Constraint satisfaction problems are ubiquitous in several different areas of artificialintelligence and computer science, because constraints are widely used to specify
Periodic Constraint Satisfaction Problems: PolynomialTime Algorithms
 In Proc. of Int. Conf. on Principles and Practice of Constraint Programming, LNCS 2833
, 2003
"... We study a generalization of the constraint satisfaction problem (CSP), the periodic constraint satisfaction problem. An input instance of the periodic CSP is a finite set of "generating" constraints over a structured variable set that implicitly specifies a larger, possibly infinite s ..."
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Cited by 3 (0 self)
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We study a generalization of the constraint satisfaction problem (CSP), the periodic constraint satisfaction problem. An input instance of the periodic CSP is a finite set of "generating" constraints over a structured variable set that implicitly specifies a larger, possibly infinite set of constraints; the problem is to decide whether or not the larger set of constraints has a satisfying assignment. This model is natural for studying constraint networks consisting of constraints obeying a high degree of regularity or symmetry. Our main contribution is the identification of two broad polynomialtime tractable subclasses of the periodic CSP.
Constraint Satisfaction: A Personal Perspective
"... Attempts at classifying computational problems as polynomial time solvable, NPcomplete,or belonging to a higher level in the polynomial hierarchy, face the difficulty of undecidability. These classes, ..."
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Attempts at classifying computational problems as polynomial time solvable, NPcomplete,or belonging to a higher level in the polynomial hierarchy, face the difficulty of undecidability. These classes,