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Consider a multiple input-multiple output (MIMO) interference channel where each transmitter and receiver are equipped with multiple antennas. An effective approach to practically achieving high system throughput is to deploy linear transceivers (or beamformers) that can optimally exploit the spatial characteristics of the channel. The recent work of Cadambe and Jafar (IEEE Trans. Inf. Theory, vol. 54, no. 8) suggests that optimal beamformers should maximize the total degrees of freedom and achieve interference alignment in the high signal-to-noise ratio (SNR) regime. In this paper we first consider the interference alignment problem without channel extension and prove that the problem of maximizing the total achieved degrees of freedom for a given MIMO interference channel is NP-hard. Furthermore, we show that even checking the achievability of a given tuple of degrees of freedom for all receivers is NP-hard when each receiver is equipped with at least three antennas. Interestingly, the same problem becomes polynomial time solvable when each transmit/receive node is equipped with no more than two antennas. We also propose a distributed algorithm for transmit covariance matrix design that does not require the DoF tuple preassignment, under the assumption that each receiver uses a linear minimum mean square error (MMSE) beamformer. The simulation results show that the proposed algorithm outperforms the existing interference alignment algorithms in terms of system throughput.