Reduction operations in fuzzy or valued constraint satisfaction

In the constraint satisfaction problem (CSP) knowledge about a relation on n variables is given in the form of constraints on subsets of the variables. A solution to the CSP is simply an instantiation of the n variables which satisfies all the constraints. The fuzzy constraint satisfaction problem (FCSP) is a generalisation of the CSP to the case in which the constraints are not categorical but represent preferences in the form of fuzzy sets. The FCSP is then an optimisation problem in the space of all possible instantiations of the n variables. It has potentially many more applications in real-world problems than the CSP.When the aim is to maximise the value of the most violated constraint, then the FCSP can be solved by solving a logarithmic number of CSPs. Fuzzy k-consistency can even be established by an algorithm whose worst-case complexity is identical to that of an optimal k-consistency algorithm for CSPs.When the operator used to aggregate constraint values is strictly monotonic (as is the case in MAX-CSP, for example) the classic operation of arc consistency has to be completely redefined. We show that an ordered version of arc consistency exists for all FCSPs with a strictly monotonic aggregation operator, and that, in the case of MAX-CSP, it can be established by an algorithm whose worst-case complexity is identical to that of the optimal arc consistency algorithm for CSPs.Neighbourhood substitution in CSPs can be generalised to fuzzy neighbourhood substitution in FCSPs, whether the aggregation operator is strictly monotonic or idempotent. We give an algorithm for applying fuzzy neighbourhood substitutions until convergence based on the corresponding algorithm for CSPs.