A large variety of problems in Artificial Intelligence and other areas of computer science can be viewed as a special case of the constraint satisfaction problem. Some examples are machine vision, belief maintenance, scheduling, temporal reasoning, graph problems, floor plan design, planning genetic experiments, and the satisfiability problem. A number of different approaches have been developed for solving these problems. Some of them use constraint propagation to simplify the original problem. Others use backtracking to directly search for possible solutions. Some are a combination of these two techniques. This paper presents a brief overview of many of these approaches in a tutorial fashion.

. Constraint satisfaction problems have wide application in artificial intelligence. They involve finding values for problem variables where the values must be consistent in that they satisfy restrictions on which combinations of values are allowed. Two standard techniques used in solving such problems are backtrack search and consistency inference. Conventional wisdom in the constraint satisfaction community suggests: 1) using consistency inference as preprocessing before search to prune values from consideration reduces subsequent search effort and 2) using consistency inference during search to prune values from consideration is best done at the limited level embodied in the forward checking algorithm. We present evidence contradicting both pieces of conventional wisdom, and suggesting renewed consideration of an approach which fully maintains arc consistency during backtrack search. 1 Introduction Constraint satisfaction problems (CSPs) involve finding values for prob...

Constraint networks are used more and more to solve combinatorial problems in real-life applications. Much activity is concentrated on improving the efficiency of finding a solution in a constraint network (the constraint satisfaction problem, CSP). Particularly, arc consistency caught many researchers' attention, involving the discovery of a large number of algorithms. And, for the last two years, it has been shown that maintaining arc consistency during search is a worthwhile approach. However, results on CSPs and on arc consistency are almost always limited to binary constraint networks. The CSP is no longer an academic problem, and it is time to deal with non-binary CSPs, as widely required in real world constraint solvers. This paper proposes a general schema to implement arc consistency on constraints of any arity when no specific algorithm is known. A first instantiation of the schema is presented here, which deals with constraints given by a predicate, by the set of forbidden c...

It is well known that any non-binary discrete constraint satisfaction problem (CSP) can be translated into an equivalent binary CSP. Two translations are known: the dual graph translation and the hidden variable translation. However, there has been little theoretical or experimental work on how well backtracking algorithms perform on these binary representations in comparison to their performance on the corresponding non-binary CSP. We present both theoretical and empirical results to help understand the tradeoffs involved. In particular, we show that translating a non-binary CSP into a binary representation can be a viable solution technique in certain circumstances. The ultimate aim of this research is to give guidance for when one should consider translating between non-binary and binary representations. Our results supply some initial answers to this question.

Solving non-binary constraint satisfaction problems, a crucial challenge for the next years, can be tackled in two different ways: translating the non-binary problem into an equivalent binary one, or extending binary search algorithms to solve directly the original problem. The latter option raises some issues when we want to extend denitions written for the binary case. This paper focuses on the well-known forward checking algorithm, and shows that it can be generalized to several non-binary versions, all tting its binary denition. The classical version, proposed by Van Hentenryck, is only one of these generalizations.

Local consistency has proven to be an important concept in the theory and practice of constraint networks. In this paper, we present a new definition of local consistency, called relational consistency. The new definition is relation-based, in contrast with the previous definition of local consistency, which we characterize as variable-based. We show the conceptual power of the new definition by showing how it unifies known elimination operators such as resolution in theorem proving, joins in relational databases, and variable elimination for solving linear inequalities. Algorithms for enforcing various levels of relational consistency are introduced and analyzed. We also show the usefulness of the new definition in characterizing relationships between properties of constraint networks and the level of local consistency needed to ensure global consistency.

The use of constraint propagation is the main feature of any constraint solver. It is thus of

Arc consistency algorithms are used in solving constraint satisfaction problems and are important in constraint logic programming languages. Search order heuristics for arc consistency algorithms significantly enhance the efficiency of their implementation. In this paper we propose and evaluate several ordering heuristics. Care is taken with experimental design, involving random problems, and statistical evaluation of results. A heuristic is identified which yields about 50% savings on average, using the standard measure of consistency pair checks, with reasonable heuristic computation cost. 1 Introduction Arc consistency insures that any two mutually constraining problem variables are mutually consistent: given a value for one, we can find a value for the other which satisfies the constraint between them. The constraint specifies which pairs of values can be simultaneously assumed by the pair of variables. Arc consistency is a fundamental concept in constraint-based reasoning [ Mack...

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In recent years, many improvements to backtracking algorithms for solving constraint satisfaction problems have been proposed. The techniques for improving backtracking algorithms can be conveniently classified as look-ahead schemes and look-back schemes. Unfortunately, look-ahead and look-back schemes are not entirely orthogonal as it has been observed empirically that the enhancement of look-ahead techniques is sometimes counter-productive to the effects of look-back techniques. In this paper, we focus on the relationship between the two most important look-ahead techniques -- using a variable ordering heuristic and maintaining a level of local consistency during the backtracking search -- and the look-back technique of conflict-directed backjumping (CBJ). We show that there exists a "perfect" dynamic variable ordering such that CBJ becomes redundant. We also show theoretically that as the level of local consistency that is maintained in the backtracking search is increased, the less that backjumping will be an improvement. Our theoretical results partially explain why a backtracking algorithm doing more in the look-ahead phase cannot benefit more from the backjumping look-back scheme. Finally, we show empirically that adding CBJ to a backtracking algorithm that maintains generalized arc consistency (GAC), an algorithm that we refer to as GAC-CBJ, can still provide orders of magnitude speedups. Our empirical results contrast with Bessiere and Regin's conclusion (1996) that CBJ is useless to an algorithm that maintains arc consistency.