Boolean constraint satisfaction: complexity results for optimization problems with arbitrary weights

A boolean constraint satisfaction problem consists of some finite set of constraints (i.e., functions from 0/1-vectors to {0, 1}) and an instance of such a problem is a set of constraints applied to specified subsets of n boolean variables. The goal is to find an assignment to the variables which satisfy all constraint applications. The computational complexity of optimization problems in connection with such problems has been studied extensively but the results have relied on the assumption that the weights are non-negative. The goal of this article is to study variants of these optimization problems where arbitrary weights are allowed. For the four problems that we consider, we give necessary and sufficient conditions for when the problems can be solved in polynomial time. In addition, we show that the problems are NP-equivalent in all other cases. (C) 2000 Elsevier Science B.V. All rights reserved.