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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
Watched literals for constraint propagation in minion
 In Proc. CP2006, 182–197
, 2006
"... Abstract. Efficient constraint propagation is crucial to any constraint solver. We show that watched literals, already a great success in the propositional satisfiability community, can also be used to provide highly efficient implementations of constraint propagators. We describe in detail three im ..."
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Cited by 43 (15 self)
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Abstract. Efficient constraint propagation is crucial to any constraint solver. We show that watched literals, already a great success in the propositional satisfiability community, can also be used to provide highly efficient implementations of constraint propagators. We describe in detail three important aspects of watched literals as we apply them to constraints, and we describe how they are implemented in the Minion constraint solver. We show three successful applications of watched literals to constraint propagators: the sum of boolean variables; generalised arc consistency for the ‘element ’ constraint; and generalised arc consistency for the ‘table ’ constraint. 1
Propositional Satisfiability and Constraint Programming: a Comparative Survey
 ACM Computing Surveys
, 2006
"... Propositional Satisfiability (SAT) and Constraint Programming (CP) have developed as two relatively independent threads of research, crossfertilising occasionally. These two approaches to problem solving have a lot in common, as evidenced by similar ideas underlying the branch and prune algorithms ..."
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Cited by 38 (4 self)
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Propositional Satisfiability (SAT) and Constraint Programming (CP) have developed as two relatively independent threads of research, crossfertilising occasionally. These two approaches to problem solving have a lot in common, as evidenced by similar ideas underlying the branch and prune algorithms that are most successful at solving both kinds of problems. They also exhibit differences in the way they are used to state and solve problems, since SAT’s approach is in general a blackbox approach, while CP aims at being tunable and programmable. This survey overviews the two areas in a comparative way, emphasising the similarities and differences between the two and the points where we feel that one technology can benefit from ideas or experience acquired
A study of residual supports in arc consistency
 In Proceedings of IJCAI’07
, 2007
"... Abstract. In an Arc Consistency (AC) algorithm, a residual support, or residue, is a support that has been stored during a previous execution of the procedure which determines if a value is supported by a constraint. The point is that a residue is not guaranteed to represent a lower bound of the sma ..."
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Cited by 32 (17 self)
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Abstract. In an Arc Consistency (AC) algorithm, a residual support, or residue, is a support that has been stored during a previous execution of the procedure which determines if a value is supported by a constraint. The point is that a residue is not guaranteed to represent a lower bound of the smallest current support of a value. In this paper, we study the theoretical impact of exploiting residues with respect to the basic algorithm AC3. First, we prove that AC3r (AC3 with unidirectional residues) and AC3rm (AC3 with multidirectional residues) are optimal for low and high constraint tightness. Second, we show that when AC has to be maintained during a backtracking search (the wellknown MAC algorithm), MAC2001 presents, with respect to MAC3r and MAC3rm, an overhead in O(µed) per branch of the binary tree built by MAC, where µ denotes the number of refutations of the branch, e the number of constraints and d the greatest domain size of the constraint network. One consequence is that, MAC3r and MAC3rm admit a better worstcase time complexity than MAC2001 for a branch involving µ refutations when either µ> d 2 or µ> d and the tightness of any constraint is either low or high. Our experimental results clearly show that exploiting residues allows enhancing MAC and SAC algorithms. 1
Data structures for generalised arc consistency for extensional constraints
 In Proceedings of the Twenty Second Conference on Artificial Intelligence
, 2007
"... Extensional (table) constraints are an important tool for attacking combinatorial problems with constraint programming. Recently there has been renewed interest in fast propagation algorithms for these constraints. We describe the use of two alternative data structures for maintaining generalised ar ..."
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Cited by 32 (9 self)
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Extensional (table) constraints are an important tool for attacking combinatorial problems with constraint programming. Recently there has been renewed interest in fast propagation algorithms for these constraints. We describe the use of two alternative data structures for maintaining generalised arc consistency on extensional constraints. The first, the NextDifference list, is novel and has been developed with this application in mind. The second, the trie, is well known but its use in this context is novel. Empirical analyses demonstrate the efficiency of the resulting approaches, both in GACschema, and in the watchedliteral table constraint in Minion.
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
Domain filtering consistencies for nonbinary constraints
 ARTIFICIAL INTELLIGENCE
, 2008
"... In nonbinary constraint satisfaction problems, the study of local consistencies that only prune values from domains has so far been largely limited to generalized arc consistency or weaker local consistency properties. This is in contrast with binary constraints where numerous such domain filtering ..."
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Cited by 26 (9 self)
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In nonbinary constraint satisfaction problems, the study of local consistencies that only prune values from domains has so far been largely limited to generalized arc consistency or weaker local consistency properties. This is in contrast with binary constraints where numerous such domain filtering consistencies have been proposed. In this paper we present a detailed theoretical, algorithmic and empirical study of domain filtering consistencies for nonbinary problems. We study three domain filtering consistencies that are inspired by corresponding variable based domain filtering consistencies for binary problems. These consistencies are stronger than generalized arc consistency, but weaker than pairwise consistency, which is a strong consistency that removes tuples from constraint relations. Among other theoretical results, and contrary to expectations, we prove that these new consistencies do not reduce to the variable based definitions of their counterparts on binary constraints. We propose a number of algorithms to achieve the three consistencies. One of these algorithms has a time complexity comparable to that for generalized arc consistency despite performing more pruning. Experiments demonstrate that our new consistencies are promising as they can be more efficient than generalized arc consistency on certain nonbinary problems.
Optimization of simple tabular reduction for table constraints
 In Proceedings of CP’08
, 2008
"... Abstract. Table constraints play an important role within constraint programming. Recently, many schemes or algorithms have been proposed to propagate table constraints or/and to compress their representation. We show that simple tabular reduction (STR), a technique proposed by J. Ullmann to dynamic ..."
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Cited by 25 (12 self)
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Abstract. Table constraints play an important role within constraint programming. Recently, many schemes or algorithms have been proposed to propagate table constraints or/and to compress their representation. We show that simple tabular reduction (STR), a technique proposed by J. Ullmann to dynamically maintain the tables of supports, is very often the most efficient practical approach to enforce generalized arc consistency within MAC. We also describe an optimization of STR which allows limiting the number of operations related to validity checking or search of supports. Interestingly enough, this optimization makes STR potentially r times faster where r is the arity of the constraint(s). The results of an extensive experimentation that we have conducted with respect to random and structured instances indicate that the optimized algorithm we propose is usually around twice as fast as the original STR and can be up to one order of magnitude faster than previous stateoftheart algorithms on some series of instances. 1
Random constraint satisfaction: easy generation of hard (satisfiable) instances
 Artificial Intelligence
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