Concept

# Constraint satisfaction

Summary
In artificial intelligence and operations research, constraint satisfaction is the process of finding a solution through a set of constraints that impose conditions that the variables must satisfy. A solution is therefore a set of values for the variables that satisfies all constraints—that is, a point in the feasible region. The techniques used in constraint satisfaction depend on the kind of constraints being considered. Often used are constraints on a finite domain, to the point that constraint satisfaction problems are typically identified with problems based on constraints on a finite domain. Such problems are usually solved via search, in particular a form of backtracking or local search. Constraint propagation are other methods used on such problems; most of them are incomplete in general, that is, they may solve the problem or prove it unsatisfiable, but not always. Constraint propagation methods are also used in conjunction with search to make a given problem simpler to solve. Other considered kinds of constraints are on real or rational numbers; solving problems on these constraints is done via variable elimination or the simplex algorithm. Constraint satisfaction as a general problem originated in the field of artificial intelligence in the 1970s (see for example ). However, when the constraints are expressed as multivariate linear equations defining (in)equalities, the field goes back to Joseph Fourier in the 19th century: George Dantzig's invention of the Simplex Algorithm for Linear Programming (a special case of mathematical optimization) in 1946 has allowed determining feasible solutions to problems containing hundreds of variables. During the 1980s and 1990s, embedding of constraints into a programming language were developed. The first languages devised expressly with intrinsic support for constraint programming was Prolog. Since then, constraint-programming libraries have become available in other languages, such as C++ or Java (e.g., Choco for Java).
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