On the Mixed-CSP problem

In the classical Constraint Satisfaction Problem(CSP) two finite models are given and we are asked to find their homomorphism. In the Mixed-CSP problem, besides the models, a set of weighted pairs of elements of this two models is given and the task is to find a homomorphism that maximizes the weight of pairs consistent with the homomorphism, i.e. pairs for which homomorphism maps the first element of the pair to the second element. Mixed-CSP can be considered as a generic model for a class of combinatorial optimization problems, one of which is a maximal independent set. It appears naturally in supervised learning when we are posed a problem of finding a function satisfying some constraints and minimizing the error on a training set. This problem shares a lot of common with the classical CSP. We show that it allows similar algebraic approach to the classification of tractable cases of this problem that connects it with relational and functional clones of multi-valued logic. In the boolean case complete classification was obtained. In general case, classes of order and conservative arithmetical predicates was introduced and it was shown that these classes are efficiently solvable.