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37
Singleton Consistencies
, 2000
"... We perform a comprehensive theoretical and empirical study of the bene ts of singleton consistencies. Our theoretical results help place singleton consistencies within the hierarchy of local consistencies. To determine the practical value of these theoretical results, we measured the costeffectiven ..."
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Cited by 48 (9 self)
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We perform a comprehensive theoretical and empirical study of the bene ts of singleton consistencies. Our theoretical results help place singleton consistencies within the hierarchy of local consistencies. To determine the practical value of these theoretical results, we measured the costeffectiveness of preprocessing with singleton consistency algorithms. Our experiments use both random and structured problems. Whilst preprocessing with singleton consistencies is not in general beneficial for random problems, it starts to pay off when randomness and structure are combined, and it is very worthwhile with structured problems like Golomb rulers. On such problems, preprocessing with consistency techniques as strong as singleton generalized arcconsistency (the singleton extension of generalized arcconsistency) can reduce runtimes. We also show that limiting algorithms that enforce singleton consistencies to a single pass often gives a small reduction in the amount of pruning and improves their costeffectiveness. These experimental results also demonstrate that conclusions from studies on random problems should be treated with caution.
Encodings of NonBinary Constraint Satisfaction Problems
, 1999
"... We perform a detailed theoretical and empirical comparison of the dual and hidden variable encodings of nonbinary constraint satisfaction problems. We identify a simple relationship between the two encodings by showing how we can translate between the two by composing or decomposing relations. ..."
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Cited by 46 (9 self)
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We perform a detailed theoretical and empirical comparison of the dual and hidden variable encodings of nonbinary constraint satisfaction problems. We identify a simple relationship between the two encodings by showing how we can translate between the two by composing or decomposing relations. This translation
Function Variables for Constraint Programming
, 2003
"... We introduce function variables to constraint programs (CP), variables whose values are one of (exponentially many) possible functions between two sets. Such variables are useful for modelling problems from domains such as configuration, planning, scheduling, etc. We show that a function variable ca ..."
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Cited by 42 (5 self)
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We introduce function variables to constraint programs (CP), variables whose values are one of (exponentially many) possible functions between two sets. Such variables are useful for modelling problems from domains such as configuration, planning, scheduling, etc. We show that a function variable can be mapped into different representations in terms of integer and set variables, and illustrate how to map constraints stated on a function variable into constraints on integer and set variables. As a result, a constraint model expressed using function variables allows for the generation of alternate CP models. Furthermore, we present an extensive theoretical comparison of models of problems involving injective functions supported by asymptotic and empirical studies. Finally, we present and evaluate a practical modelling tool that is based on a highlevel language that supports function variables. The tool helps users explore different alternate CP models starting from a function model that is easy to develop, understand, and maintain.
A fast and simple algorithm for bounds consistency of the alldifferent constraint
 IN PROCEEDINGS OF THE 18TH INTERNATIONAL JOINT CONFERENCE ON ARTIFICIAL INTELLIGENCE
, 2003
"... In constraint programming one models a problem by stating constraints on acceptable solutions. The constraint model is then usually solved by interleaving backtracking search and constraint propagation. Previous studies have demonstrated that designing special purpose constraint propagators for comm ..."
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Cited by 42 (9 self)
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In constraint programming one models a problem by stating constraints on acceptable solutions. The constraint model is then usually solved by interleaving backtracking search and constraint propagation. Previous studies have demonstrated that designing special purpose constraint propagators for commonly occurring constraints can significantly improve the efficiency of a constraint programming approach. In this paper we present a fast, simple algorithm for bounds consistency propagation of the alldifferent constraint. The algorithm has the same worst case behavior as the previous best algorithm but is much faster in practice. Using a variety of benchmark and random problems, we show that our algorithm outperforms existing bounds consistency algorithms and also outperforms—on problems with an easily identifiable property—stateoftheart commercial implementations of propagators for stronger forms of local consistency.
Permutation Problems and Channelling Constraints
 TR 26, APES Group
, 2001
"... When writing a constraint program, we have to decide what to make the decision variable, and how to represent the constraints on these variables. In many cases, there is considerable choice for the decision variables. For example, with permutation problems, we can choose between a primal and a dual ..."
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Cited by 36 (1 self)
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When writing a constraint program, we have to decide what to make the decision variable, and how to represent the constraints on these variables. In many cases, there is considerable choice for the decision variables. For example, with permutation problems, we can choose between a primal and a dual representation. In the dual representation, dual variables stand for the primal values, whilst dual values stand for the primal variables. By means of channelling constraints, a combined model can have both primal and dual variables. In this paper, we perform an extensive theoretical and empirical study of these different models. Our results will aid constraint programmers to choose a model for a permutation problem. They also illustrate a general methodology for comparing different constraint models.
A Tutorial on Constraint Programming
 University of Leeds
, 1995
"... A constraint satisfaction problem (CSP) consists of a set of variables; for each variable, a finite set of possible values (its domain); and a set of constraints restricting the values that the variables can simultaneously take. A solution to a CSP is an assignment of a value from its domain to ever ..."
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Cited by 35 (3 self)
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A constraint satisfaction problem (CSP) consists of a set of variables; for each variable, a finite set of possible values (its domain); and a set of constraints restricting the values that the variables can simultaneously take. A solution to a CSP is an assignment of a value from its domain to every variable, in such a way that every constraint is satisfied. Many problems arising in O.R., in particular scheduling, timetabling and other combinatorial problems, can be represented as CSPs. Constraint programming tools now exist which allow CSPs to be expressed easily, and provide standard strategies for finding solutions. This tutorial is intended to give a basic grounding in constraint satisfaction problems and some of the algorithms used to solve them, including the techniques commonly used in constraint programming tools. In particular, it covers arc and path consistency; simple backtracking and forward checking, as examples of search algorithms; and the use of heuristics to guide the...
CGRASS: A System for Transforming Constraint Satisfaction Problems
 Recent Advances in Constraints, 1530, LNCS 2627
, 2002
"... Abstract. Experts at modelling constraint satisfaction problems (CSPs) carefully choose model transformations to reduce greatly the amount of effort that is required to solve a problem by systematic search. It is a considerable challenge to automate such transformations and to identify which transfo ..."
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Cited by 32 (9 self)
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Abstract. Experts at modelling constraint satisfaction problems (CSPs) carefully choose model transformations to reduce greatly the amount of effort that is required to solve a problem by systematic search. It is a considerable challenge to automate such transformations and to identify which transformations are useful. Transformations include adding constraints that are implied by other constraints, adding constraints that eliminate symmetrical solutions, removing redundant constraints and replacing constraints with their logical equivalents. This paper describes the CGRASS (Constraint Generation And Symmetrybreaking) system that can improve a problem model by automatically performing transformations of these kinds. We focus here on transforming individual CSP instances. Experiments on the Golomb ruler problem suggest that producing good problem formulations solely by transforming problem instances is, generally, infeasible. We argue that, in certain cases, it is better to transform the problem class than individual instances and, furthermore, it can sometimes be better to transform formulations of a problem that are more abstract than a CSP. 1
Dual Modelling of Permutation and Injection Problems
 Journal of Artificial Intelligence Research
, 2004
"... When writing a constraint program, we have to choose which variables should be the decision variables, and how to represent the constraints on these variables. In many cases, there is considerable choice for the decision variables. Consider, for example, permutation problems in which we have as many ..."
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Cited by 31 (9 self)
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When writing a constraint program, we have to choose which variables should be the decision variables, and how to represent the constraints on these variables. In many cases, there is considerable choice for the decision variables. Consider, for example, permutation problems in which we have as many values as variables, and each variable takes an unique value. In such problems, we can choose between a primal and a dual viewpoint. In the dual viewpoint, each dual variable represents one of the primal values, whilst each dual value represents one of the primal variables. Alternatively, by means of channelling constraints to link the primal and dual variables, we can have a combined model with both sets of variables. In this paper, we perform an extensive theoretical and empirical study of such primal, dual and combined models for two classes of problems: permutation problems and injection problems. Our results show that it often be advantageous to use multiple viewpoints, and to have constraints which channel between them to maintain consistency. They also illustrate a general...
Automated reformulation of specifications by safe delay of constraints
 Proceedings of the 2nd International Workshop on Modelling and Reformulating Constraint Satisfaction Problems
, 2003
"... In this paper we propose a form of reasoning on specifications of combinatorial problems, with the goal of reformulating them so that they are more efficiently solvable. The reformulation technique highlights constraints that can be safely “delayed”, and solved afterwards. Our main contribution is t ..."
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Cited by 19 (8 self)
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In this paper we propose a form of reasoning on specifications of combinatorial problems, with the goal of reformulating them so that they are more efficiently solvable. The reformulation technique highlights constraints that can be safely “delayed”, and solved afterwards. Our main contribution is the characterization (with soundness proof) of safedelay constraints with respect to a criterion on the specification, thus obtaining a mechanism for the automated reformulation of specifications applicable to a great variety of problems, e.g., graph coloring and jobshop scheduling. This is an advancement with respect to the forms of reasoning done by stateoftheartsystems, which typically just detect linearity of specifications. Another contribution is a preliminary experimentation on the effectiveness of the proposed technique, which reveals promising time savings.
Theory and practice of constraint propagation
 In Proceedings of the 3rd Workshop on Constraint Programming in Decision and Control
, 2001
"... Abstract: Despite successful application of constraint programming (CP) to solving many reallife problems there is still an indispensable group or researchers considering (wrongly) CP as a simple evaluation technique only. Even if sophisticated search algorithms play an important role in solving co ..."
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Cited by 19 (2 self)
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Abstract: Despite successful application of constraint programming (CP) to solving many reallife problems there is still an indispensable group or researchers considering (wrongly) CP as a simple evaluation technique only. Even if sophisticated search algorithms play an important role in solving constraintbased models, the real power engine behind CP is called constraint propagation (domain filtering, pruning or consistency techniques). In the paper we give a survey of common consistency techniques for binary constraints. We describe the main ideas behind them, list their advantages and limitations, and compare their pruning power. Then we briefly explain how these techniques can be extended to nonbinary constraints. Last part of the paper is devoted to modelling issues. We give some hints how the constraint propagation can be exploited more when solving reallife problems. This part is based on our experience with solving reallife programs and it is also supported by empirical observations of other researchers.