In this paper we study the complexity of the weighted maximum constraint satisfaction problem (
) over an arbitrary finite domain. In this problem, one is given a collection of weighted constraints on overlapping sets of variables, and the goal is to find an assignment of values to the variables so as to maximize the total weight of satisfied constraints.
-hard in general; however, some restrictions on the form of constraints may ensure tractability. Recent results indicate that there is a connection between tractability of such restricted problems and supermodularity of the allowed constraint types with respect to some lattice ordering of the domain. We prove several results confirming this in a special case when the lattice ordering is as loose as possible, i.e., a diamond one.