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.

We introduce a general framework for constraint satisfaction and optimization where classical CSPs, fuzzy CSPs, weighted CSPs, partial constraint satisfaction, and others can be easily cast. The framework is based on a semiring structure, where the set of the semiring specifies the values to be associated with each tuple of values of the variable domain, and the two semiring operations (1 and 3) model constraint projection and combination respectively. Local consistency algorithms, as usually used for classical CSPs, can be exploited in this general framework as well, provided that certain conditions on the semiring operations are satisfied. We then show how this framework can be used to model both old and new constraint solving and optimization schemes, thus allowing one to both formally justify many informally taken choices in existing schemes, and to prove that local consistency techniques can be used also in newly defined schemes.

Many combinatorial search problems can be expressed as `constraint satisfaction problems', and this class of problems is known to be NP-complete in general. In this paper we investigate the subclasses which arise from restricting the possible constraint types. We first show that any set of constraints which does not give rise to an NP-complete class of problems must satisfy a certain type of algebraic closure condition. We then investigate all the different possible forms of this algebraic closure property, and establish which of these are sufficient to ensure tractability. As examples, we show that all known classes of tractable constraints over finite domains can be characterised by such an algebraic closure property. Finally, we describe a simple computational procedure which can be used to determine the closure properties of a given set of constraints. This procedure involves solving a particular constraint satisfaction problem, which we call an `indicator problem'. Keywords: Cons...

We introduce a general framework for constraint solving where classical CSPs, fuzzy CSPs, weighted CSPs, partial constraint satisfaction, and others can be easily cast. The framework is based on a semiring structure, where the set of the semiring specifies the values to be associated to each tuple of values of the variable domain, and the two semiring operations (+ and x) model constraint projection and combination respectively. Local consistency algorithms, as usually used for classical CSPs, can be exploited in this general framework as well, provided that some conditions on the semiring operations are satisfied. We then show how this framework can be used to model both old and new constraint solving schemes, thus allowing one both to formally justify many informally taken choices in existing schemes, and to prove that the local consistency techniques can be used also in newly defined schemes. 1

In this paper we describe and compare two frameworks for constraint solving where classical CSPs, fuzzy CSPs, weighted CSPs, partial constraint satisfaction, and others can be easily cast. One is based on a semiring, and the other one on a totally ordered commutative monoid. While comparing the two approaches, we show how to pass from one to the other one, and we discuss when this is possible. The two frameworks have been independently introduced in [2], [3] and [35].

A solution of a Constraint Satisfaction Problem (CSP) is an assignment of values to all its variables such that all its constraints are satisfied. Usually two CSPs are considered equivalent if they have the same solution set. We find this definition limiting, and develop a more general definition based on the concept of mutual reducibility. In this extended scheme it is reasonable to consider a pair of CSPs equivalent even if they have different solutions. The basic idea behind the extended scheme is that two CSPs can be considered equivalent whenever they contain the same "amount of information", i.e. whenever it is possible to obtain the solution of one of them from the solution of the other one, and viceversa. In this way, both constraint and variable redundancy are allowed in CSPs belonging to the same equivalence class. As an example of the usefulness of this new notion of equivalence, we formally prove that binary and non-binary CSPs are equivalent (in the new sense). Such a pro...

. We show that several constraint propagation algorithms (also called (local) consistency, consistency enforcing, Waltz, filtering or narrowing algorithms) are instances of algorithms that deal with chaotic iteration. To this end we propose a simple abstract framework that allows us to classify and compare these algorithms and to establish in a uniform way their basic properties. Note. This is a full, revised version of our article "From Chaotic Iteration to Constraint Propagation", Proc. of 24th International Colloquium on Automata, Languages and Programming (ICALP '97), (invited lecture) , Springer-Verlag Lecture Notes in Computer Science 1256, pp. 3655, (1997). Keywords: constraint propagation, chaotic iteration, generic algorithms. 1 Introduction 1.1 Motivation Over the last ten years constraint programming emerged as an interesting and viable approach to programming. In this approach the programming process is limited to a generation of requirements ("constraints") and a solut...

Finding solutions to a binary constraint satisfaction problem is known to be an NP-complete problem in general, but may be tractable in cases where either the set of allowed constraints or the graph structure is restricted. This paper considers restricted sets of contraints which are closed under permutation of the labels. We identify a set of constraints which gives rise to a class of tractable problems and give polynomial time algorithms for solving such problems, and for finding the equivalent minimal network. We also prove that the class of problems generated by any set of constraints not contained in this restricted set is NP-complete. 1 Introduction Finding solutions to a constraint satisfaction problem is known to be an NPcomplete problem in general [11] even when the constraints are restricted to binary constraints. However, many of the problems which arise in practice have special properties which allow them to be solved efficiently. The question of identifying restrictions t...

Finding solutions to a constraint satisfaction problem is known to be an NP-complete problem in general, but may be tractable in cases where either the set of allowed constraints or the graph structure is restricted. In this paper we identify a restricted set of contraints which gives rise to a class of tractable problems. This class generalizes the notion of a Horn formula in propositional logic to larger domain sizes. We give a polynomial time algorithm for solving such problems, and prove that the class of problems generated by any larger set of constraints is NP-complete. 1 Introduction Combinatorial problems abound in Artificial Intelligence. Examples include planning, temporal reasoning, line-drawing labelling and circuit design. The Constraint Satisfaction Problem (CSP) [14] is a generic combinatorial problem which is widely studied in the AI community because it allows all of these problems to be expressed in a natural and direct way. Reduction operations [12, 10] and intellig...

Although the constraint satisfaction problem is NP-complete in general, a number of constraint classes have been identified for which some fixed level of local consistency is sufficient to ensure global consistency. In this paper, we describe a simple algebraic property which characterises all possible constraint types for which strong k-consistency is sufficient to ensure global consistency, for each k ? 2. We give a number of examples to illustrate the application of this result. 1 Introduction The constraint satisfaction problem provides a framework in which it is possible to express, in a natural way, many combinatorial problems encountered in artificial intelligence and elsewhere. The aim in a constraint satisfaction problem is to find an assignment of values to a given set of variables subject to constraints on the values which can be assigned simultaneously to certain specified subsets of variables. The constraint satisfaction problem is known to be an NP-complete problem in ge...