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Our attention in this paper is focused on radio resource allocation (RRA) problems in orthogonal frequency division multiple access (OFDMA) systems. By assuming perfect channel estimation for all users, a fast optimal algorithm is developed to solve two classes of RRA problems: one class is to minimize the total transmission power at base station under the quality of service (QoS) constraint of each user, and the other class is to achieve maximum system data throughput under the constraints of maximal transmission power at base station and QoS of each user. The proposed algorithm is developed on the basis of the well-known branch-and-bound method. As shown in our results, the proposed algorithm offers the same performance as the optimal one achieved by using exhaustive full-search algorithm. However, the computational complexity involved in the proposed algorithm is significantly reduced.