Because of its ease of administration, role-based access control (RBAC) has become the norm to enforcing security in most of today's organizations. For implementing RBAC, it is important to devise a complete and correct set of roles. This task, known as role engineering, has been identified as one of the costliest components in deploying RBAC. A key problem with respect to role engineering is that there is no formal metric for measuring the goodness/interestingness of the devised set of roles. Recently, Vaidya et al. [26], formally define the role mining problem (RMP) as the problem of discovering an optimal set of roles from existing user permissions, and analyze its theoretical bounds. Essentially, given a user-permission assignment (UPA), the basic RMP is to discover the user-role assignment relation (UA) and role-permission assignment relation (PA) such that the number of roles required is minimum. In this paper, we present another interesting and useful problem, called the edge-RMP, with a different minimality objective. The edge-RMP, requires the discovery of a complete and correct set of roles such that the discovered |UA|+|PA| is the minimum possible. Minimal |UA|+|PA| is a useful metric as it would minimize the administrative burden since less number of assignments need to be managed. Although the basic-RMP and the edge-RMP appear to be related problems, we demonstrate with concrete examples that they are, in fact, independent of each other. We prove that the edge-RMP is an NP-hard problem by reducing the known “vertex cover problem” to the decision version of the edge-RMP. Another important contribution of this paper is to provide a binary integer programming solution to this problem by showing that the edge-RMP can be formulated in that form. As a result, one can directly borrow existing implementation solutions for binary integer programming and guide further research in this direction. We also propose a heuristic solution for large scale problems, and experimentally validate our algorithm.