The complexity of maximal constraint languages

Many combinatorial search problems can be expressed as “constraint satisfaction problems” using an appropriate “constraint language”, that is, a set of relations over some fixed finite set of values. It is well-known that there is a trade-off between the expressive power of a constraint language and the complexity of the problems it can express. In the present paper we systematically study the complexity of all maximal constraint languages, that is, languages whose expressive power is just weaker than that of the language of all constraints. Using the algebraic invariance properties of constraints, we exhibit a strong necessary condition for tractability of such a constraint language. Moreover, we show that, at least for small sets of values, this condition is also sufficient.

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