Maximum Constraint Satisfaction on Diamonds

In this paper we study the complexity of the (weighted) maximum constraint satisfaction problem (Max CSP) 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. Max Cut is a typical example of a Max CSP problem. Max CSP is NP-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. Diamonds are the smallest lattices in terms of the number of comparabilities, and so are as unordered as a lattice can possibly be. In the present paper, we study Max CSP on diamond-ordered domains. We show that if all allowed constraints are supermodular with respect to such an ordering then the problem can be solved in polynomial (in fact, in cubic) time. We also prove a partial converse: if the set of allowed constraints includes a certain small family of binary supermodular constraints on such a lattice, then the problem is tractable if and only if all of the allowed constraints are supermodular; otherwise, it is NP-hard.