Abstract. Interest in role-based access control has generated considerable research activity in recent years. A number of interesting problems related to the well known set cover problem have come to light as a result of this activity. However, the computational complexity of some of these problems is still not known. In this paper, we explore some variations on the set cover problem and use these variations to establish the computational complexity of these problems. Most significantly, we introduce the minimal cover problem ? a generalization of the set cover problem ? which we use to determine the complexity of the inter-domain role mapping problem.