In orthogonal frequency division multiplexing systems, reduced feedback schemes provide essential channel state information from the users to the base station (BS) without overwhelming the uplink. For the practically important best-M scheme, in which each user feeds back only its M strongest subchannels and their indices to the BS, we derive a novel, throughput-optimal scheduling and rate adaptation policy that enables the BS to schedule the best user and its data rate for all the subchannels. The policy exploits the structure of the information fed back by the best-M scheme and the correlation among subchannel gains. We present it in closed-form for the widely studied exponential correlation model. Using insights gleaned from the optimal policy, we propose a novel, low-complexity two subchannel reduction approach, which is seen empirically to be near-optimal and easily handles practically important general channel correlation models, quantized feedback, and co-channel interference in multi-cell scenarios. Compared with several ad hoc approaches, the proposed approaches improve the cell throughput without any additional feedback. A modified gradient-based opportunistic scheduler is also proposed to ensure user fairness. © 2017 IEEE.