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Generalized Arc Consistency for Global Cardinality Constraint
"... A global cardinality constraint (gcc) is specified in terms of a set of variables X = fx1 ; :::; xpg which take their values in a subset of V = fv1 ; :::; vdg. It constrains the number of times a value v i 2 V is assigned toavariable in X to be in an interval (l i ;c i ). Cardinality constraints hav ..."
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Cited by 207 (10 self)
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A global cardinality constraint (gcc) is specified in terms of a set of variables X = fx1 ; :::; xpg which take their values in a subset of V = fv1 ; :::; vdg. It constrains the number of times a value v i 2 V is assigned toavariable in X to be in an interval (l i ;c i ). Cardinality constraints have proved very useful in many reallife problems, suchas scheduling, timetabling, or resource allocation. A gcc is more general than a constraint of difference, which requires each interval to be #0; 1#. In this paper, we present an efficient way of implementing generalized arc consistency for a gcc. The algorithm we propose is based on a new theorem of flow theory. Its space complexity is O(#Xj#jVj) and its time complexity is O(jXj 2 #jVj). We also show how this algorithm can efficiently be combined with other filtering techniques.
Arc Consistency for General Constraint Networks: Preliminary Results
, 1997
"... Constraint networks are used more and more to solve combinatorial problems in reallife applications. Much activity is concentrated on improving the efficiency of finding a solution in a constraint network (the constraint satisfaction problem, CSP). Particularly, arc consistency caught many research ..."
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Cited by 139 (16 self)
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Constraint networks are used more and more to solve combinatorial problems in reallife applications. Much activity is concentrated on improving the efficiency of finding a solution in a constraint network (the constraint satisfaction problem, CSP). Particularly, arc consistency caught many researchers' attention, involving the discovery of a large number of algorithms. And, for the last two years, it has been shown that maintaining arc consistency during search is a worthwhile approach. However, results on CSPs and on arc consistency are almost always limited to binary constraint networks. The CSP is no longer an academic problem, and it is time to deal with nonbinary CSPs, as widely required in real world constraint solvers. This paper proposes a general schema to implement arc consistency on constraints of any arity when no specific algorithm is known. A first instantiation of the schema is presented here, which deals with constraints given by a predicate, by the set of forbidden c...
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.
Radio Link Frequency Assignment
 Constraints
, 1999
"... The problem of radio frequency assignment is to provide communication channels from limited spectral resources whilst keeping to a minimum the interference suffered by those whishing to communicate in a given radio communication network. This problem is a combinatorial (NPhard) optimization problem ..."
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Cited by 70 (11 self)
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The problem of radio frequency assignment is to provide communication channels from limited spectral resources whilst keeping to a minimum the interference suffered by those whishing to communicate in a given radio communication network. This problem is a combinatorial (NPhard) optimization problem. In 1993, the CELAR (the French "Centre d'Electronique de l'Armement") built a suite of simplified versions of Radio Link Frequency Assignment Problems (RLFAP) starting from data on a real network (Roisnel 93). Initially designed for assessing the performances of several Constraint Logic Programming languages, these benchmarks have been made available to the public in the framework of the European EUCLID project CALMA (Combinatorial Algorithms for Military Applications).
Using Constraint Metaknowledge to Reduce Arc Consistency Computation
 Artificial Intelligence
, 1999
"... Constraint satisfaction problems are widely used in articial intelligence. They involve nding values for problem variables subject to constraints that specify which combinations of values are consistent. Knowledge about properties of the constraints can permit inferences that reduce the cost of cons ..."
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Cited by 62 (8 self)
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Constraint satisfaction problems are widely used in articial intelligence. They involve nding values for problem variables subject to constraints that specify which combinations of values are consistent. Knowledge about properties of the constraints can permit inferences that reduce the cost of consistency checking. In particular, such inferences can be used to reduce the number of constraint checks required in establishing arc consistency, a fundamental constraintbased reasoning technique. A general ACInference algorithm schema is presented and various forms of inference discussed. A specific algorithm, AC7, is presented, which takes advantage of a simple property common to all binary constraints to eliminate constraint checks that other arc consistency algorithms perform. The effectiveness of this approach is demonstrated analytically, and experimentally.
MAC and Combined Heuristics: Two Reasons to Forsake FC (and CBJ?) on Hard Problems
 In Proceedings of the Second International Conference on Principles and Practice of Constraint Programming
, 1996
"... . In the last twenty years, many algorithms and heuristics were developed to find solutions in constraint networks. Their number increased to such an extent that it quickly became necessary to compare their performances in order to propose a small number of "good" methods. These comparison ..."
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Cited by 49 (3 self)
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. In the last twenty years, many algorithms and heuristics were developed to find solutions in constraint networks. Their number increased to such an extent that it quickly became necessary to compare their performances in order to propose a small number of "good" methods. These comparisons often led us to consider FC or FCCBJ associated with a "minimum domain" variable ordering heuristic as the best techniques to solve a wide variety of constraint networks. In this paper, we first try to convince once and for all the CSP community that MAC is not only more efficient than FC to solve large practical problems, but it is also really more efficient than FC on hard and large random problems. Afterwards, we introduce an original and efficient way to combine variable ordering heuristics. Finally, we conjecture that when a good variable ordering heuristic is used, CBJ becomes an expensive gadget which almost always slows down the search, even if it saves a few constraint checks. 1 Introducti...
From Restricted Path Consistency to MaxRestricted Path Consistency
 PROCEEDINGS OF THIRD INTERNATIONAL CONFERENCE ON PRINCIPLES AND PRACTICE OF CONSTRAINT PROGRAMMING (CP97
, 1997
"... There is no need to show the importance of the filtering techniques to solve constraint satisfaction problems i.e. to find values for problem variables subject to constraints that specify whichcombinations of values are consistent. They can be used during a preprocessingstep to remove once and for a ..."
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Cited by 46 (13 self)
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There is no need to show the importance of the filtering techniques to solve constraint satisfaction problems i.e. to find values for problem variables subject to constraints that specify whichcombinations of values are consistent. They can be used during a preprocessingstep to remove once and for all some local inconsistencies, or duringthesearch toefficiently prunethe search tree. Recently, in [5], a comparison of the most practicable filteringtechniques concludes that restricted pathconsistency (RPC) is a promising local consistency that requires little additional cpu time compared to arc consistency while removing most of thepathinverse inconsistentvalues. However, the RPC algorithm used for this comparison (presented in [1] and called RPC1 in the following) has a non optimal worst case time complexity and bad average timeand space complexities. Therefore, we propose RPC2, a new RPC algorithm with O(end 2 )worst case time complexity and requiring less space than RPC1 in practice. The second aim of this paper is to extend RPC tonew local consistencies, kRPC and MaxRPC, andto compare their pruning efficiency withtheother practicable local consistencies. Furthermore, we propose andstudy a MaxRPC algorithm based on AC6 thatwe used for this comparison.