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Partial Constraint Satisfaction
, 1992
"... . A constraint satisfaction problem involves finding values for variables subject to constraints on which combinations of values are allowed. In some cases it may be impossible or impractical to solve these problems completely. We may seek to partially solve the problem, in particular by satisfying ..."
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Cited by 471 (21 self)
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. A constraint satisfaction problem involves finding values for variables subject to constraints on which combinations of values are allowed. In some cases it may be impossible or impractical to solve these problems completely. We may seek to partially solve the problem, in particular by satisfying a maximal number of constraints. Standard backtracking and local consistency techniques for solving constraint satisfaction problems can be adapted to cope with, and take advantage of, the differences between partial and complete constraint satisfaction. Extensive experimentation on maximal satisfaction problems illuminates the relative and absolute effectiveness of these methods. A general model of partial constraint satisfaction is proposed. 1 Introduction Constraint satisfaction involves finding values for problem variables subject to constraints on acceptable combinations of values. Constraint satisfaction has wide application in artificial intelligence, in areas ranging from temporal r...
Breaking row and column symmetries in matrix models
 Proceedings of Eighth International Conference on Principles and Practice of Constraint Programming (CPO2
, 2002
"... Abstract. We identify an important class of symmetries in constraint programming, arising from matrices of decision variables where rows and columns can be swapped. Whilst lexicographically ordering the rows (columns) breaks all the row (column) symmetries, lexicographically ordering both the rows ..."
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Cited by 115 (37 self)
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Abstract. We identify an important class of symmetries in constraint programming, arising from matrices of decision variables where rows and columns can be swapped. Whilst lexicographically ordering the rows (columns) breaks all the row (column) symmetries, lexicographically ordering both the rows and the columns fails to break all the compositions of the row and column symmetries. Nevertheless, our experimental results show that this is effective at dealing with these compositions of symmetries. We extend these results to cope with symmetries in any number of dimensions, with partial symmetries, and with symmetric values. Finally, we identify special cases where all compositions of the row and column symmetries can be eliminated by the addition of only a linear number of symmetrybreaking constraints. 1
Minion: A fast scalable constraint solver
 In: Proceedings of ECAI 2006, Riva del Garda
, 2006
"... Abstract. We present Minion, a new constraint solver. Empirical results on standard benchmarks show orders of magnitude performance gains over stateoftheart constraint toolkits. These gains increase with problem size – Minion delivers scalable constraint solving. Minion is a generalpurpose const ..."
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Cited by 114 (38 self)
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Abstract. We present Minion, a new constraint solver. Empirical results on standard benchmarks show orders of magnitude performance gains over stateoftheart constraint toolkits. These gains increase with problem size – Minion delivers scalable constraint solving. Minion is a generalpurpose constraint solver, with an expressive input language based on the common constraint modelling device of matrix models. Focussing on matrix models supports a highlyoptimised implementation, exploiting the properties of modern processors. This contrasts with current constraint toolkits, which, in order to provide ever more modelling and solving options, have become progressively more complex at the cost of both performance and usability. Minion is a black box from the user point of view, deliberately providing few options. This, combined with its raw speed, makes Minion a substantial step towards Puget’s ‘Model and Run ’ constraint solving paradigm. 1
The Complexity of Global Constraints
, 2004
"... We study the computational complexity of reasoning with global constraints. We show that reasoning with such constraints is intractable in general. We then demonstrate how the same tools of computational complexity can be used in the design and analysis of specific global constraints. In particular, ..."
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Cited by 82 (23 self)
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We study the computational complexity of reasoning with global constraints. We show that reasoning with such constraints is intractable in general. We then demonstrate how the same tools of computational complexity can be used in the design and analysis of specific global constraints. In particular, we illustrate how computational complexity can be used to determine when a lesser level of local consistency should be enforced, when decomposing constraints will lose pruning, and when combining constraints is tractable. We also show how the same tools can be used to study symmetry breaking, metaconstraints like the cardinality constraint, and learning nogoods.
Deriving Filtering Algorithms from Constraint Checkers
 Principles and Practice of Constraint Programming (CP’2004), volume 3258 of LNCS
, 2004
"... Abstract. This reportdeals with global constraints for which the set of solutions can be recognized by an extended finite automaton whose size is bounded by a polynomial in ¦ , where ¦ is the number of variables of the corresponding global constraint. By reformulating the automaton as a conjunction ..."
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Cited by 51 (11 self)
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Abstract. This reportdeals with global constraints for which the set of solutions can be recognized by an extended finite automaton whose size is bounded by a polynomial in ¦ , where ¦ is the number of variables of the corresponding global constraint. By reformulating the automaton as a conjunction of signature and transition constraints we show how to systematically obtain a filtering algorithm. Under some restrictions on the signature and transition constraints this filtering algorithm achieves arcconsistency. An implementation based on some constraints as well as on the metaprogramming facilities of SICStus Prolog is available. For a restricted class of automata we provide a filtering algorithm for the relaxed case, where the violation cost is the minimum number of variables to unassign in order to get back to a solution. Keywords: Constraint Programming,
ArcConsistency for a Chain of Lexicographic Ordering Constraints
, 2002
"... We present an arcconsistency algorithm for a chain of lexicographic ordering constraints on m vectors of n variables each. The algorithm maintains arcconsistency and runs in O(nmd) time per invocation, where d is the cost of certain domain operations. ..."
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Cited by 34 (1 self)
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We present an arcconsistency algorithm for a chain of lexicographic ordering constraints on m vectors of n variables each. The algorithm maintains arcconsistency and runs in O(nmd) time per invocation, where d is the cost of certain domain operations.
Global constraints for integer and set value precedence
 In: Proceedings of 10th International Conference on Principles and Practice of Constraint Programming (CP2004
, 2004
"... Abstract. The paper introduces value precedence on integer and set sequences. A useful application of the notion is in breaking symmetries of indistinguishable values, an important class of symmetries in practice. Although value precedence can be expressed straightforwardly using ifthen constraint ..."
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Cited by 30 (2 self)
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Abstract. The paper introduces value precedence on integer and set sequences. A useful application of the notion is in breaking symmetries of indistinguishable values, an important class of symmetries in practice. Although value precedence can be expressed straightforwardly using ifthen constraints in existing constraint programming systems, the resulting formulation is inefficient both in terms of size and runtime. We present two propagation algorithms for implementing global constraints on value precedence in the integer and set domains. Besides conducting experiments to verify the feasibility and efficiency of our proposal, we characterize also the propagation level attained by various usages of the global constraints as well as the conditions when the constraints can be used consistently with other types of symmetry breaking constraints.
General symmetry breaking constraints
 In: 12th International Conference on Principles and Practices of Constraint Programming (CP2006), SpringerVerlag
, 2006
"... Abstract. We present some general constraints for breaking symmetries in constraint satisfaction problems. These constraints can be used to break symmetries acting on variables, values, or both. We also consider symmetry breaking constraints to deal with conditional symmetries, and symmetries acting ..."
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Cited by 28 (16 self)
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Abstract. We present some general constraints for breaking symmetries in constraint satisfaction problems. These constraints can be used to break symmetries acting on variables, values, or both. We also consider symmetry breaking constraints to deal with conditional symmetries, and symmetries acting on set and other types of variables. 1
Constraint Models for the Covering Test Problem
, 2006
"... Covering arrays can be applied to the testing of software, hardware and advanced materials, and to the effects of hormone interaction on gene expression. In this paper we develop constraint programming models of the problem of finding an optimal covering array. Our models exploit global constraints ..."
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Cited by 25 (1 self)
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Covering arrays can be applied to the testing of software, hardware and advanced materials, and to the effects of hormone interaction on gene expression. In this paper we develop constraint programming models of the problem of finding an optimal covering array. Our models exploit global constraints, multiple viewpoints and symmetrybreaking constraints. We show that compound variables, representing tuples of variables in our original model, allow the constraints of this problem to be represented more easily and hence propagate better. With our best integrated model, we are able to either prove the optimality of existing bounds or find new optimal solutions, for arrays of moderate size. Local search on a SATencoding of the model is able to find improved solutions and bounds for larger problems.
Propagation algorithms for lexicographic ordering constraints
 Artificial Intelligence
, 2006
"... Finitedomain constraint programming has been used with great success to tackle a wide variety of combinatorial problems in industry and academia. To apply finitedomain constraint programming to a problem, it is modelled by a set of constraints on a set of decision variables. A common modelling pat ..."
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Cited by 24 (8 self)
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Finitedomain constraint programming has been used with great success to tackle a wide variety of combinatorial problems in industry and academia. To apply finitedomain constraint programming to a problem, it is modelled by a set of constraints on a set of decision variables. A common modelling pattern is the use of matrices of decision variables. The rows and/or columns of these matrices are often symmetric, leading to redundancy in a systematic search for solutions. An effective method of breaking this symmetry is to constrain the assignments of the affected rows and columns to be ordered lexicographically. This paper develops an incremental propagation algorithm, GACLexLeq, that establishes generalised arc consistency on this constraint in O(n) operations, where n is the length of the vectors. Furthermore, this paper shows that decomposing GACLexLeq into primitive constraints available in current finitedomain constraint toolkits reduces the strength or increases the cost of constraint propagation. Also presented are extensions and modifications to the algorithm to handle strict lexicographic ordering, detection of entailment, and vectors of unequal length. Experimental results on a number of domains demonstrate the value of GACLexLeq. 1