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61
Refining the basic constraint propagation algorithm
 In Proceedings IJCAI’01
, 2001
"... Propagating constraints is the main feature of any constraint solver. This is thus of prime importance to manage constraint propagation as efficiently as possible, justifying the use of the best algorithms. But the ease of integration is also one of the concerns when implementing an algorithm in a c ..."
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Cited by 100 (12 self)
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Propagating constraints is the main feature of any constraint solver. This is thus of prime importance to manage constraint propagation as efficiently as possible, justifying the use of the best algorithms. But the ease of integration is also one of the concerns when implementing an algorithm in a constraint solver. This paper focuses on AC3, which is the simplest arc consistency algorithm known so far. We propose two refinements that preserve as much as possible the ease of integration into a solver (no heavy data structure to be maintained during search), while giving some noticeable improvements in efficiency. One of the proposed refinements is analytically compared to AC6, showing interesting properties, such as optimality of its worstcase time complexity. 1
An Optimal Coarsegrained Arc Consistency Algorithm
 Artificial Intelligence
"... The use of constraint propagation is the main feature of any constraint solver. It is thus of prime importance to manage the propagation in an efficient and effective fashion. There are two classes of propagation algorithms for general constraints: finegrained algorithms where the removal of a val ..."
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Cited by 93 (16 self)
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The use of constraint propagation is the main feature of any constraint solver. It is thus of prime importance to manage the propagation in an efficient and effective fashion. There are two classes of propagation algorithms for general constraints: finegrained algorithms where the removal of a value for a variable will be propagated to the corresponding values for other variables, and coarsegrained algorithms where the removal of a value will be propagated to the related variables. One big advantage of coarsegrained algorithms, like AC3, over finegrained algorithms, like AC4, is the ease of integration when implementing an algorithm in a constraint solver. However, finegrained algorithms usually have optimal worst case time complexity while coarsegrained algorithms don’t. For example, AC3 is an algorithm with nonoptimal worst case complexity although it is simple, efficient in practice, and widely used. In this paper we propose a coarsegrained algorithm, AC2001/3.1, that is worst case optimal and preserves as much as possible the ease of its integration into a solver (no heavy data structure to be maintained during search). Experimental results show that AC2001/3.1 is competitive with the best finegrained algorithms such as AC6. The idea behind the new algorithm can immediately be applied to obtain a path consistency algorithm that has the bestknown time and space complexity. The same idea is then extended to nonbinary constraints. Preliminary versions of this paper appeared in [BR01, ZY01].
Constraint propagation
 Handbook of Constraint Programming
, 2006
"... Constraint propagation is a form of inference, not search, and as such is more ”satisfying”, both technically and aesthetically. —E.C. Freuder, 2005. Constraint reasoning involves various types of techniques to tackle the inherent ..."
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Cited by 77 (5 self)
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Constraint propagation is a form of inference, not search, and as such is more ”satisfying”, both technically and aesthetically. —E.C. Freuder, 2005. Constraint reasoning involves various types of techniques to tackle the inherent
Domain Filtering Consistencies
 Journal of Artificial Intelligence Research (JAIR)
, 2001
"... Enforcing local consistencies is one of the main features of constraint reasoning. Which level of local consistency should be used when searching for solutions in a constraint network is a basic question. Arc consistency and partial forms of arc consistency have been widely studied, and have been kn ..."
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Cited by 74 (8 self)
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Enforcing local consistencies is one of the main features of constraint reasoning. Which level of local consistency should be used when searching for solutions in a constraint network is a basic question. Arc consistency and partial forms of arc consistency have been widely studied, and have been known for sometime through the forward checking or the MAC search algorithms. Until recently, stronger forms of local consistency remained limited to those that change the structure of the constraint graph, and thus, could not be used in practice, especially on large networks. This paper focuses on the local consistencies that are stronger than arc consistency, without changing the structure of the network, i.e., only removing inconsistent values from the domains. In the last five years, several such local consistencies have been proposed by us or by others. We make an overview of all of them, and highlight some relations between them. We compare them both theoretically and experimentally, considering their pruning efficiency and the time required to enforce them.
Backjumpbased backtracking for constraint satisfaction problems
 Artificial Intelligence
"... The performance of backtracking algorithms for solving nitedomain constraint satisfaction problems can be improved substantially by lookback and lookahead methods. Lookback techniques extract information by analyzing failing search paths that are terminated by deadends. Lookahead techniques ..."
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Cited by 44 (2 self)
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The performance of backtracking algorithms for solving nitedomain constraint satisfaction problems can be improved substantially by lookback and lookahead methods. Lookback techniques extract information by analyzing failing search paths that are terminated by deadends. Lookahead techniques use constraint propagation algorithms to avoid such deadends altogether. This survey describes a number of lookback variants including backjumping and constraint recording which recognize and avoid some unnecessary explorations of the search space. The last portion of the paper gives an overview of lookahead methods such as forward checking and dynamic variable ordering, and discusses their combination with backjumping.
Theoretical analysis of singleton arc consistency
 Proceedings ECAI’04 Workshop on Modelling and solving problems with constraints
, 2004
"... Singleton arc consistency (SAC) is a consistency property that is simple to specify and is stronger than arc consistency. Algorithms have already been proposed to enforce SAC, but they have a high time complexity. In this paper, we give a lower bound to the worstcase time complexity of enforcing SA ..."
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Cited by 29 (3 self)
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Singleton arc consistency (SAC) is a consistency property that is simple to specify and is stronger than arc consistency. Algorithms have already been proposed to enforce SAC, but they have a high time complexity. In this paper, we give a lower bound to the worstcase time complexity of enforcing SAC on binary constraints. We discuss two interesting features of SAC. The first feature leads us to propose an algorithm for SAC that has optimal time complexity when restricted to binary constraints. The second feature leads us to extend SAC to a stronger level of local consistency that we call Bidirectional SAC (BiSAC). We also show the relationship between SAC and the propagation of disjunctive constraints. 1
Generalized arc consistency for positive table constraints
 In Proceedings of CP’06
, 2006
"... Abstract. In this paper, we propose a new algorithm to establish Generalized Arc Consistency (GAC) on positive table constraints, i.e. constraints defined in extension by a set of allowed tuples. Our algorithm visits the lists of valid and allowed tuples in an alternative fashion when looking for a ..."
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Cited by 27 (10 self)
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Abstract. In this paper, we propose a new algorithm to establish Generalized Arc Consistency (GAC) on positive table constraints, i.e. constraints defined in extension by a set of allowed tuples. Our algorithm visits the lists of valid and allowed tuples in an alternative fashion when looking for a support (i.e. a tuple that is both allowed and valid). It is then able to jump over sequences of valid tuples containing no allowed tuple and over sequences of allowed tuples containing no valid tuple. Our approach, that can be easily grafted to any generic GAC algorithm, admits on some instances a behaviour quadratic in the arity of the constraints whereas classical approaches, i.e. approaches that focus on either valid or allowed tuples, admit an exponential behaviour. We show the effectiveness of this approach, both theoretically and experimentally. 1
Exploiting multidirectionality in coarsegrained arc consistency algorithms
 In Proc. of CP’03
, 2003
"... Abstract. Arc consistency plays a central role in solving Constraint Satisfaction Problems. This is the reason why many algorithms have been proposed to establish it. Recently, an algorithm called AC2001 and AC3.1 has been independently presented by their authors. This algorithm which is considered ..."
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Cited by 23 (12 self)
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Abstract. Arc consistency plays a central role in solving Constraint Satisfaction Problems. This is the reason why many algorithms have been proposed to establish it. Recently, an algorithm called AC2001 and AC3.1 has been independently presented by their authors. This algorithm which is considered as a refinement of the basic algorithm AC3 has the advantage of being simple and competitive. However, it does not take into account constraint bidirectionality as AC7 does. In this paper, we address this issue, and, in particular, introduce two new algorithms called AC3.2 and AC3.3 which benefit from good properties of both AC3 and AC7. Indeed, AC3.2 and AC3.3 are as easy to implement as AC3 and take advantage of bidirectionality as AC7 does. More precisely, AC3.2 is a general algorithm which partially exploits bidirectionality whereas AC3.3 is a binary algorithm which fully exploits bidirectionality. It turns out that, when Maintaining Arc Consistency during search, MAC3.2, due to a memorization effect, is more efficient than MAC3.3 both in terms of constraint checks and cpu time. Compared to MAC2001/3.1, our experimental results show that MAC3.2 saves about 50% of constraint checks and, on average, 15 % of cpu time. 1
A fast arc consistency algorithm for nary constraints
 In Proceedings of AAAI’05
, 2005
"... The GACScheme has become a popular general purpose algorithm for solving nary constraints, although it may scan an exponential number of supporting tuples. In this paper, we develop a major improvement of this scheme. When searching for a support, our new algorithm is able to skip over a number ..."
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Cited by 21 (1 self)
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The GACScheme has become a popular general purpose algorithm for solving nary constraints, although it may scan an exponential number of supporting tuples. In this paper, we develop a major improvement of this scheme. When searching for a support, our new algorithm is able to skip over a number of tuples exponential in the arity of the constraint by exploiting knowledge about the current domains of the variables. We demonstrate the effectiveness of the method for large table constraints.