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This paper addresses the problem of frequency domain packet scheduling (FDPS) incorporating spatial division multiplexing (SDM) multiple input multiple output (MIMO) techniques on the 3GPP Long-Term Evolution (LTE) downlink. We impose the LTE MIMO constraint of selecting only one MIMO mode (spatial multiplexing or transmit diversity) per user per transmission time interval (TTI). First, we address the optimal MIMO mode selection (multiplexing or diversity) per user in each TTI in order to maximize the proportional fair (PF) criterion adapted to the additional frequency and spatial domains. We prove that both single-user (SU-) and multi-user (MU-) MIMO FDPS problems under the LTE requirement are NP-hard. We therefore develop two types of approximation algorithms (ones with full channel feedback and the others with partial channel feedback), all of which guarantee provable performance bounds for both SU- and MU-MIMO cases. Based on 3GPP LTE system model simulations, our approximation algorithms that take into account both spatial and frequency diversity gains outperform the exact algorithms that do not exploit the potential spatial diversity gain. Moreover, the approximation algorithms with partial channel feedback achieve comparable performance (with only 1-6 percent performance degradation) to the ones with full channel feedback, while significantly reducing the channel feedback overhead by nearly 50 percent.