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49
Adaptive Constraint Satisfaction
 WORKSHOP OF THE UK PLANNING AND SCHEDULING
, 1996
"... Many different approaches have been applied to constraint satisfaction. These range from complete backtracking algorithms to sophisticated distributed configurations. However, most research effort in the field of constraint satisfaction algorithms has concentrated on the use of a single algorithm fo ..."
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Cited by 919 (43 self)
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Many different approaches have been applied to constraint satisfaction. These range from complete backtracking algorithms to sophisticated distributed configurations. However, most research effort in the field of constraint satisfaction algorithms has concentrated on the use of a single algorithm for solving all problems. At the same time, a consensus appears to have developed to the effect that it is unlikely that any single algorithm is always the best choice for all classes of problem. In this paper we argue that an adaptive approach should play an important part in constraint satisfaction. This approach relaxes the commitment to using a single algorithm once search commences. As a result, we claim that it is possible to undertake a more focused approach to problem solving, allowing for the correction of bad algorithm choices and for capitalising on opportunities for gain by dynamically changing to more suitable candidates.
A Theoretical Evaluation of Selected Backtracking Algorithms
 Artificial Intelligence
, 1997
"... In recent years, many new backtracking algorithms for solving constraint satisfaction problems have been proposed. The algorithms are usually evaluated by empirical testing. This method, however, has its limitations. Our paper adopts a di erent, purely theoretical approach, which is based on charact ..."
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Cited by 121 (2 self)
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In recent years, many new backtracking algorithms for solving constraint satisfaction problems have been proposed. The algorithms are usually evaluated by empirical testing. This method, however, has its limitations. Our paper adopts a di erent, purely theoretical approach, which is based on characterizations of the sets of search treenodes visited by the backtracking algorithms. A notion of inconsistency between instantiations and variables is introduced, and is shown to be a useful tool for characterizing such wellknown concepts as backtrack, backjump, and domain annihilation. The characterizations enable us to: (a) prove the correctness of the algorithms, and (b) partially order the algorithms according to two standard performance measures: the number of nodes visited, and the number of consistency checks performed. Among other results, we prove the correctness of Backjumping and Con ictDirected Backjumping, and show that Forward Checking never visits more nodes than Backjumping. Our approach leads us also to propose a modi cation to two hybrid backtracking algorithms, Backmarking with Backjumping (BMJ) and Backmarking with Con ictDirected Backjumping (BMCBJ), so that they always perform fewer consistency checks than the original algorithms. 1
Exploiting symmetries within constraint satisfaction search
, 2001
"... Symmetry often appears in realworld constraint satisfaction problems, but strategies for exploiting it are only beginning to be developed. Here, a framework for exploiting symmetry within depthfirst search is proposed, leading to two heuristics for variable selection and a domain pruning procedure ..."
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Cited by 43 (1 self)
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Symmetry often appears in realworld constraint satisfaction problems, but strategies for exploiting it are only beginning to be developed. Here, a framework for exploiting symmetry within depthfirst search is proposed, leading to two heuristics for variable selection and a domain pruning procedure. These strategies are then applied to two highly symmetric combinatorial problems, namely the Ramsey problem and the generation of balanced incomplete block designs. Experimental results show that these generalpurpose strategies can compete with, and in some cases outperform, previous more ad hoc procedures.
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 40 (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 Polynomial Time Algorithm for the NQueens Problem
, 1990
"... this paper we present a new, probabilistic local search algorithm which is based on a gradientbased heuristic. This efficient algorithm is capable of finding a solution for extremely large size nqueens problems. We give the execution statistics for this algorithm with n up to 500,000. Keywords: Ar ..."
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Cited by 35 (4 self)
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this paper we present a new, probabilistic local search algorithm which is based on a gradientbased heuristic. This efficient algorithm is capable of finding a solution for extremely large size nqueens problems. We give the execution statistics for this algorithm with n up to 500,000. Keywords: Artificial intelligence (AI), combinatorial search, gradientbased heuristic, local search, the nqueens problem, nonbacktracking search, fast search algorithm. 1
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 33 (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...
Interchangeability Supports Abstraction and Reformulation for Constraint Satisfaction
 In Proceedings of Symposium on Abstraction, Reformulation and Approximation (SARA'95
, 1995
"... Abstraction and reformulation are fundamental, powerful ideas in artificial intelligence, but they have not had a great deal of application in the area of constraint satisfaction. The obvious way to implement abstraction in a constraint satisfaction context is to simplify the problem by removing con ..."
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Cited by 31 (1 self)
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Abstraction and reformulation are fundamental, powerful ideas in artificial intelligence, but they have not had a great deal of application in the area of constraint satisfaction. The obvious way to implement abstraction in a constraint satisfaction context is to simplify the problem by removing constraints, and then use the solutions to the simplified problem to guide the search for a solution to the original problem. In a sense, local search, hill climbing methods use this approach, while giving up completeness guarantees. The problem for a complete method is that a simplified problem may be trivial (recent work on locating hard problems suggests that hard problems may cluster on narrow ridges in problem space). Thus there may be too many solutions to the simplified problem to be useful. The Cartesian product representation and interchangeability techniques provide ways of working with compact representations of large sets of solutions.
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...
Binary vs. nonbinary constraints
 Artificial Intelligence
, 2002
"... Fellowship program. 1 There are two well known transformations from nonbinary constraints to binary constraints applicable to constraint satisfaction problems (CSPs) with finite domains: the dual transformation and the hidden (variable) transformation. We perform a detailed formal comparison of the ..."
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Cited by 30 (3 self)
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Fellowship program. 1 There are two well known transformations from nonbinary constraints to binary constraints applicable to constraint satisfaction problems (CSPs) with finite domains: the dual transformation and the hidden (variable) transformation. We perform a detailed formal comparison of these two transformations. Our comparison focuses on two backtracking algorithms that maintain a local consistency property at each node in their search tree: the forward checking and maintaining arc consistency algorithms. We first compare local consistency techniques such as arc consistency in terms of their inferential power when they are applied to the original (nonbinary) formulation and to each of its binary transformations. For example, we prove that enforcing arc consistency on the original formulation is equivalent to enforcing it on the hidden transformation. We then extend these results to the two backtracking algorithms. We are able to give either a theoretical bound on how much one formulation is better than another, or examples that show such a bound does not exist. For example, we prove that the performance of the forward checking algorithm applied to the hidden transformation of a problem is within a polynomial bound of the performance of the same algorithm applied to the dual transformation of the problem. Our results can be used to help decide if applying one of these transformations to all (or part) of a constraint satisfaction model would be beneficial. 2 1
Using CBR to select solution strategies in constraint programming
 In ICCBR
, 2005
"... Abstract. Constraint programming is a powerful paradigm that offers many different strategies for solving problems. Choosing a good strategy is difficult; choosing a poor strategy wastes resources and may result in a problem going unsolved. We show how CaseBased Reasoning can be used to select good ..."
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Cited by 24 (0 self)
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Abstract. Constraint programming is a powerful paradigm that offers many different strategies for solving problems. Choosing a good strategy is difficult; choosing a poor strategy wastes resources and may result in a problem going unsolved. We show how CaseBased Reasoning can be used to select good strategies. We design experiments which demonstrate that, on two problems with quite different characteristics, CBR can outperform four other strategy selection techniques. 1