This paper addresses the downlink resource allocation problem in pre‐5G (LTE‐B) networks. At each time slot, the problem is to share the radio resources between users in order to maximize a given objective function. We expressed this problem considering the LTE standard constraints, which are rarely considered in the literature. Mainly, for any given user, the base station is constrained to transmit with a single modulation and coding scheme. We show that this problem is non‐deterministic polynomial‐time hard, and therefore, we propose a generic approximation algorithm that covers a large variety of objectives. This algorithm is composed of three routines that enable an effective resource sharing procedure. The first routine computes a solution for a relaxation of our main problem, while the second routine selects the most suitable modulation and coding scheme for each user. Finally, the last routine effectively distributes the unallocated resources. For the max rate policy, simulation results show that our algorithm outperforms other existing algorithms in terms of capacity and remains close to the optimal. Under the proportional fairness policy, our solution also provides a very good fairness while maintaining a near‐optimal capacity. Copyright © 2016 John Wiley & Sons, Ltd.