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In this paper, we address the problem of allocating users to radio resources (i.e. subcarriers) in the downlink of an OFDMA system. In particular, we consider a multi-format resource allocation problem (MF-RAP) in which the link adaptation adjusts the spectral efficiency for each user-subcarrier pair, i.e. for each radio link, in order to minimize the total transmission power while fulfilling a rate request for each user. We propose an integer linear programming (ILP) formulation of the problem and exhaustively discuss the computational complexity. Specifically, we prove that the problem is NP-hard in the strong sense and demonstrate that it is hard to be approximated in polynomial time within a constant factor. Hence, we present heuristic approaches that achieve “reasonably good” solutions in the general case. Computational experiences show that, in comparison with a commercial state-of-the-art ILP optimization solver, the proposed algorithms are effective in terms of solution quality and CPU times.