Tags:constraint satisfaction, polymorphism, redundancy and Turan number

Abstract:

A constraint language G has non-redundancy f(n) if every instance of CSP(G) with n variables contains at most f(n) non-redundant constraints. If G has maximum arity r then it has non-redundancy O(n^r), but there are notable examples for which this upper bound is far from the best possible. In general, the non-redundancy of constraint languages is poorly understood and little is known beyond the trivial bounds Omega(n) and O(n^r).

In this paper, we introduce an elementary algebraic framework dedicated to the analysis of the non-redundancy of constraint languages. This framework relates redundancy-preserving reductions between constraint languages to closure operators known as pattern partial polymorphisms, which can be interpreted as generic mechanisms to generate redundant constraints in CSP instances. We illustrate the power of this framework by deriving a simple characterisation of languages of arity r having non-redundancy Theta(n^r).