The user authorization query (UAQ) problem determines whether there exists an optimum set of roles to be activated to provide a set of permissions requested by a user. It has been deemed as a key issue for efficiently handling user's access requests in role-based access control (RBAC). Unfortunately, the weight is a value attached to a permission/role representing its importance, should be introduced to UAQ, has been ignored. In this paper, we propose a comprehensive definition of the weighted UAQ (WUAQ) problem with the role-weighted-cardinality and permission-weighted-cardinality constraints. Moreover, we study the computational complexity of different subcases of WUAQ, and show that many instances in each subcase are intractable. In particular, inspired by the idea of the genetic algorithm, we propose an algorithm to approximate solve an intractable subcase of the WUAQ problem. An important observation is that this algorithm can be efficiently modified to handle the other subcases of the WUAQ problem. The experimental results show the advantage of the proposed algorithm, which is especially fit for the case that the computational overhead is even more important than the accuracy in a large-scale RBAC system.