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 networks are known as a useful way to formulate problems such as design, scene labeling, temporal reasoning, and more recently natural language parsing. The problem of the existence of solutions in a constraint network is NP-complete. Hence, consistency techniques have been widely studied to simplify constraint networks before or during the search of solutions. Arc-consistency is the most used of them. Mohr and Henderson [Moh&Hen86] have proposed AC-4, an algorithm having an optimal worst-case time complexity. But it has two drawbacks: its space complexity and its average time complexity. In problems with many solutions, where the size of the constraints is large, these drawbacks become so important that users often replace AC-4 by AC-3 [Mac&Fre85], a nonoptimal algorithm. In this paper, we propose a new algorithm, AC-6, which keeps the optimal worst-case time complexity of AC-4 while working out the drawback of space complexity. More, the average time complexity of AC-6 is optimal for constraint networks where nothing is known about the semantic of the constraints. At the end of the paper, experimental results show how much AC-6 outperforms AC-3 and AC-4. 1.

Constraint programming is newly flowering in industry. Several companies have recently started up to exploit the technology, and the number of industrial applications is now growing very quickly. This survey will seek, by examples,

Learning during backtrack search is a space-intensive process that records information (such as additional constraints) in order to avoid redundant work. In this paper, we analyze the effects of polynomial-spacebounded learning on runtime complexity of backtrack search. One space-bounded learning scheme records only those constraints with limited size, and another records arbitrarily large constraints but deletes those that become irrelevant to the portion of the search space being explored. We find that relevance-bounded learning allows better runtime bounds than size-bounded learning on structurally restricted constraint satisfaction problems. Even when restricted to linear space, our relevancebounded learning algorithm has runtime complexity near that of unrestricted (exponential space-consuming) learning schemes.

Enforcing local consistencies is one of the main features of constraint reasoning. Which level of local consistency should be used when searching for solutions in a constraint network is a basic question. Arc consistency and partial forms of arc consistency have been widely studied, and have been known for sometime through the forward checking or the MAC search algorithms. Until recently, stronger forms of local consistency remained limited to those that change the structure of the constraint graph, and thus, could not be used in practice, especially on large networks. This paper focuses on the local consistencies that are stronger than arc consistency, without changing the structure of the network, i.e., only removing inconsistent values from the domains. In the last five years, several such local consistencies have been proposed by us or by others. We make an overview of all of them, and highlight some relations between them. We compare them both theoretically and experimentally, considering their pruning efficiency and the time required to enforce them.

Practical Constraint Satisfaction Problems (CSPs) such as design of integrated circuits or scheduling generally entail large search spaces with hundreds or even thousands of variables, each with hundreds or thousands of possible values. Often, only a very tiny fraction of all these possible assignments participates in a satisfactory solution. This article discusses techniques that aim at reducing the effective size of the search space to be explored in order to find a satisfactory solution by judiciously selecting the order in which variables are instantiated and the sequence in which possible values are tried for each variable. In the CSP literature, these techniques are commonly referred to as variable and value ordering heuristics. Our investigation is conducted in the job shop scheduling domain. We show that, in contrast with problems studied earlier in the CSP literature, generic variable and value heuristics do not perform well in this domain. This is attributed to the difficulty of these heuristics to properly account for the tightness of constraints and/or the connectivity of the constraint graphs induced by job shop scheduling CSPs. A new probabilistic framework is introduced that better captures these key aspects of the job shop scheduling search space. Empirical results show that variable and value ordering heuristics

Abstraction techniques are important for solving constraint satisfaction problems with global constraints and low solution density. In the presence of global constraints, backtracking search is unable to prune partial solutions. It therefore operates like pure generate-and-test. Abstraction improves on generate-and-test by enabling entire subsets of the solution space to be pruned early in a search process. This paper describes how abstraction spaces can be characterized in terms of approximate symmetries of the original, concrete search space. It defines two special types of approximate symmetry, called "range symmetry" and "domain symmetry", which apply to function finding problems. It also presents algorithms for automatically synthesizing hierarchic problem solvers based on range or domain symmetry. The algorithms operate by analyzing declarative descriptions of classes of constraint satisfaction problems. Both algorithms have been fully implemented. This paper concludes by presenting data from experiments testing the two synthesis algorithms and the resulting problem solvers on NP-hard scheduling and partitioning problems.

We propose a framework for solving CSPs based both on backtracking techniques and on the notion of tree-decomposition of the constraint networks. This mixed approach permits us to define a new framework for the enumeration, which we expect that it will benefit from the advantages of two approaches: a practical efficiency of enumerative algorithms and a warranty of a limited time complexity by an approximation of the tree-width of the constraint networks. Finally, experimental results allow us to show the advantages of this approach. 1

: Constraint programming (CP) is an emergent software technology for declarative description and effective solving of large, particularly combinatorial, problems especially in areas of planning and scheduling. It represents the most exciting developments in programming languages of the last decade and, not surprisingly, it has recently been identified by the ACM (Association for Computing Machinery) as one of the strategic directions in computer research. Not only it is based on a strong theoretical foundation but it is attracting widespread commercial interest as well, in particular, in areas of modelling heterogeneous optimisation and satisfaction problems. In the paper, we give a survey of constraint programming technology and its applications starting from the history context and interdisciplinary nature of CP. The central part of the paper is dedicated to the description of main constraint satisfaction techniques and industrial applications. We conclude with the overview of limit...

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