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34
Closure Properties of Constraints
 Journal of the ACM
, 1997
"... Many combinatorial search problems can be expressed as `constraint satisfaction problems', and this class of problems is known to be NPcomplete in general. In this paper we investigate the subclasses which arise from restricting the possible constraint types. We first show that any set of constrain ..."
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Cited by 135 (16 self)
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Many combinatorial search problems can be expressed as `constraint satisfaction problems', and this class of problems is known to be NPcomplete in general. In this paper we investigate the subclasses which arise from restricting the possible constraint types. We first show that any set of constraints which does not give rise to an NPcomplete class of problems must satisfy a certain type of algebraic closure condition. We then investigate all the different possible forms of this algebraic closure property, and establish which of these are sufficient to ensure tractability. As examples, we show that all known classes of tractable constraints over finite domains can be characterised by such an algebraic closure property. Finally, we describe a simple computational procedure which can be used to determine the closure properties of a given set of constraints. This procedure involves solving a particular constraint satisfaction problem, which we call an `indicator problem'. Keywords: Cons...
On the algebraic structure of combinatorial problems
 THEORETICAL COMPUTER SCIENCE
, 1998
"... ..."
Constraint Satisfaction Problems And Finite Algebras
, 1999
"... 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. In this paper we show that any restricted set of constraint types c ..."
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Cited by 50 (7 self)
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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. In this paper we show that any restricted set of constraint types can be associated with a finite universal algebra. We explore how the computational complexity of a restricted constraint satisfaction problem is connected to properties of the corresponding algebra. For this, we introduce a notion of `tractable algebra' and study how the tractability of an algebra relates to the tractability of its smaller derived algebras, including its subalgebras and homomorphic images. This allows us to significantly reduce the types of algebras which need to be investigated. Using these results we exhibit a common structural property of all known intractable constraint satisfaction problems. Finally, we classify all finite strictly simple surjective algebras wit...
Constraints, Consistency, and Closure
 Artificial Intelligence
, 1998
"... Although the constraint satisfaction problem is NPcomplete in general, a number of constraint classes have been identified for which some fixed level of local consistency is sufficient to ensure global consistency. In this paper, we describe a simple algebraic property which characterises all possi ..."
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Cited by 46 (12 self)
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Although the constraint satisfaction problem is NPcomplete in general, a number of constraint classes have been identified for which some fixed level of local consistency is sufficient to ensure global consistency. In this paper, we describe a simple algebraic property which characterises all possible constraint types for which strong kconsistency is sufficient to ensure global consistency, for each k ? 2. We give a number of examples to illustrate the application of this result. 1 Introduction The constraint satisfaction problem provides a framework in which it is possible to express, in a natural way, many combinatorial problems encountered in artificial intelligence and elsewhere. The aim in a constraint satisfaction problem is to find an assignment of values to a given set of variables subject to constraints on the values which can be assigned simultaneously to certain specified subsets of variables. The constraint satisfaction problem is known to be an NPcomplete problem in ge...
A Survey of Tractable Constraint Satisfaction Problems
, 1997
"... In this report we discuss constraint satisfaction problems. These are problems in which values must be assigned to a collection of variables, subject to specified constraints. We focus specifically on problems in which the domain of possible values for each variable is finite. The report surveys the ..."
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Cited by 41 (5 self)
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In this report we discuss constraint satisfaction problems. These are problems in which values must be assigned to a collection of variables, subject to specified constraints. We focus specifically on problems in which the domain of possible values for each variable is finite. The report surveys the various conditions that have been shown to be sufficient to ensure tractability in these problems. These are broken down into three categories: ffl Conditions on the overall structure; ffl Conditions on the nature of the constraints; ffl Conditions on bounded pieces of the problem. 1 Introduction A constraint satisfaction problem is a way of expressing simultaneous requirements for values of variables. The study of constraint satisfaction problems was initiated by Montanari in 1974 [34], when he used them as a way of describing certain combinatorial problems arising in imageprocessing. It was quickly realised that the same general framework was applicable to a much wider class of probl...
A Unifying Framework for Tractable Constraints
, 1995
"... . Many combinatorial search problems may be expressed as constraint satisfaction problems, and this class of problems is known to be NPcomplete in general. In this paper we examine restricted classes of constraints which lead to tractable problems. We show that all known classes with this property ..."
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Cited by 27 (12 self)
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. Many combinatorial search problems may be expressed as constraint satisfaction problems, and this class of problems is known to be NPcomplete in general. In this paper we examine restricted classes of constraints which lead to tractable problems. We show that all known classes with this property may be characterized by a simple algebraic closure condition. Using this condition provides a uniform test to establish whether a given set of constraints falls into any of the known tractable classes, and may therefore be solved efficiently. 1 Introduction Many combinatorial search problems may be expressed as constraint satisfaction problems. Unfortunately, finding solutions to a constraint satisfaction problem is known to be an NPcomplete problem in general [12] even when the constraints are restricted to binary constraints. However, many of the problems which arise in practice have special properties which allow them to be solved efficiently. The question of identifying restrictions t...
Hentenryck, Constraint satisfaction over connected row convex constraints
 in: Proc. IJCAI97
, 1997
"... This paper studies constraint satisfaction over connected rowconvex (CRC) constraints. It shows that CRC constraints are closed under composition, intersection, and transposition, the basic operations of pathconsistency algorithms. This establishes that path consistency over CRC constraints produc ..."
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Cited by 21 (0 self)
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This paper studies constraint satisfaction over connected rowconvex (CRC) constraints. It shows that CRC constraints are closed under composition, intersection, and transposition, the basic operations of pathconsistency algorithms. This establishes that path consistency over CRC constraints produces a minimal and decomposable network and is thus a polynomialtime decision procedure for CRC networks. This paper also presents a new pathconsistency algorithm for CRC constraints running in time O(n3d2) and space O(n2d), wherenisthenumber of variables and d is the size of the largest domain, improving the traditional time and space complexity by orders of magnitude. The paper also shows how to construct CRC constraints by conjunction and disjunction of a set of basic CRC constraints, highlighting how CRC constraints generalize monotone constraints and presenting interesting subclasses of CRC constraints. Experimental results show that the algorithm behaves well in practice. © 1999 Elsevier Science B.V. All rights reserved.
Tractable Disjunctive Constraints
 Journal of the ACM
, 1996
"... . Many combinatorial search problems can be expressed as `constraint satisfaction problems', and this class of problems is known to be NPcomplete in general. In this paper we investigate `disjunctive constraints', that is, constraints which have the form of the disjunction of two constraints of spe ..."
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Cited by 18 (6 self)
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. Many combinatorial search problems can be expressed as `constraint satisfaction problems', and this class of problems is known to be NPcomplete in general. In this paper we investigate `disjunctive constraints', that is, constraints which have the form of the disjunction of two constraints of specified types. We show that when the constraint types involved in the disjunction have a certain property, which we call `independence', and when a certain restricted class of problems is tractable, then the class of all problems involving these disjunctive constraints is tractable. We give examples to show that many known examples of tractable constraint classes arise in this way, and derive new tractable classes which have not previously been identified. Keywords: Constraint satisfaction problem, complexity, NPcompleteness 1 Introduction The constraint satisfaction problem provides a framework in which it is possible to express, in a natural way, many combinatorial problems encountered i...
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 language ..."
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Cited by 18 (13 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