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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.
Using Auxiliary Variables and Implied Constraints to Model NonBinary Problems
, 2000
"... We perform an extensive theoretical and empirical analysis of the use of auxiliary variables and implied constraints in modelling a class of nonbinary constraint satisfaction problems called problems of distance. This class of problems include 1d, 2d and circular Golomb rulers. We identify a larg ..."
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Cited by 38 (16 self)
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We perform an extensive theoretical and empirical analysis of the use of auxiliary variables and implied constraints in modelling a class of nonbinary constraint satisfaction problems called problems of distance. This class of problems include 1d, 2d and circular Golomb rulers. We identify a large number of different models, both binary and nonbinary, and compare theoretically the level of consistency achieved by generalized arc consistency on them. Our experiments show that the introduction of auxiliary variables and implied constraints can significantly reduce the size of the search space. For instance, our final models reduce the time to find an optimal 10mark Golomb ruler 50fold.
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...
Modelling the Golomb Ruler Problem
, 1999
"... . The Golomb ruler problem has been proposed as a challenging constraint satisfaction problem. We consider a large number of different models of this problem, both binary and nonbinary. The problem can be modelled using quaternary constraints, but in practice using a set of auxiliary variables and ..."
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Cited by 33 (9 self)
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. The Golomb ruler problem has been proposed as a challenging constraint satisfaction problem. We consider a large number of different models of this problem, both binary and nonbinary. The problem can be modelled using quaternary constraints, but in practice using a set of auxiliary variables and ternary constraints gives better results. A binary encoding of the problem gives a smaller search tree, but is impractical because it takes far longer to run. We compare variable ordering heuristics and consider the use of implied constraints to improve propagation. We believe that more case studies such as this are essential to reduce the skill currently required for successful modelling. 1 Introduction In his AAAI98 invited talk, Gene Freuder identified modelling as one of the major hurdles preventing the uptake of constraint satisfaction technology. The availability of nonbinary constraints can increase the number of possible models of a problem amnd so makes modelling still more diffi...
Symmetry Breaking as a Prelude to Implied Constraints: A Constraint Modelling Pattern
 Proc. 16th Euro. Conf. on AI, 171175
, 2004
"... Abstract. Finitedomain constraint programming can be used to solve a wide range of problems by first modelling the problem as a set of constraints that characterise the problem’s solutions, then searching for solutions that satisfy the constraints. Experts often augment models with implied constrai ..."
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Cited by 16 (7 self)
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Abstract. Finitedomain constraint programming can be used to solve a wide range of problems by first modelling the problem as a set of constraints that characterise the problem’s solutions, then searching for solutions that satisfy the constraints. Experts often augment models with implied constraints and constraints that break symmetries in the model. An emerging pattern in the modelling process, highlighted and demonstrated here, is that some powerful implied constraints can be derived only after symmetrybreaking constraints have been added. Furthermore, the choice between alternative symmetrybreaking constraints is commonly made by considering either the amount of symmetry broken or the strength of pruning obtained in comparison with the overhead of enforcing the constraints. We demonstrate that the choice should also consider the strength of the implied constraints derivable from the symmetry breaking constraints. We also discuss future automation of the selection of symmetrybreaking constraints and the derivation of implied constraints. 1
Symmetry and implied constraints in the steel mill slab design problem
 Proc. CP’01 Wshop on Modelling and Problem Formulation
, 2001
"... ..."
Scotland
"... We present a technique for using preprocessing based on mammalian early auditory processing to produce a segmentation of sound based on onsets and offsets. The sound signal is bandpassed and each band processed to enhance onsets and offsets. The onset and offset signals are compressed, then cluster ..."
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Cited by 1 (0 self)
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We present a technique for using preprocessing based on mammalian early auditory processing to produce a segmentation of sound based on onsets and offsets. The sound signal is bandpassed and each band processed to enhance onsets and offsets. The onset and offset signals are compressed, then clustered both in time and across frequency channels using a network of integrateandfire neurons. A spikebased representation of onsets and offsets is produced, and the timing of these spikes used to segment the sound. By considering spikes in varying number of bands, a multilevel segmentation tree can be built. This tree is a purely datadriven representation of the segmental structure of the sound. 1
1 Introduction Modelling for Constraint Programming
"... Constraint programming can be a successful technology ..."
Handbook of Constraint Programming 375
"... Constraint programming can be a successful technology for solving practical problems; however, there is abundant evidence that how the problem to be solved is modelled as a Constraint Satisfaction Problem (CSP) can have a dramatic effect on how easy it is to find a solution, or indeed whether it can ..."
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Constraint programming can be a successful technology for solving practical problems; however, there is abundant evidence that how the problem to be solved is modelled as a Constraint Satisfaction Problem (CSP) can have a dramatic effect on how easy it is to find a solution, or indeed whether it can realistically be solved at all. The importance of modelling in constraint programming has long been recognized e.g. in invited talks by Freuder [14] and Puget [34]. In this chapter, it will be assumed that the problem to be solved can be represented as a CSP whose domains are finite; infinite domains are discussed in Chapter 16, “Continuous and Interval Constraints”. In most of the examples, the variable domains will be sets of integers; see Chapter 17, “Constraints over Structured Domains”, for more on set variables and other variable types. A complicating factor in modelling is the interaction between the model, the search algorithm and the search heuristics. To simplify matters, it will be assumed that, having modelled the problem of interest as a CSP, the CSP will be solved using a constraint solver such as ILOG Solver, ECL i PS e, Choco, SICStus Prolog, or the like. The default complete