On the complexity of leakage interference minimization for interference alignment

For a general MIMO interference channel, we can determine the feasibility of linear interference alignment via minimizing the leakage interference. This paper gives a complete complexity characterization of the leakage interference minimization problem. It is shown that, when each transmitter (receiver) is equipped with at least three antennas and each receiver (transmitter) is equipped with at least two antennas, the problem of checking whether the interference in the network can be perfectly aligned is strongly NP-hard. Moreover, when each transmit/receive node is equipped with two or more antennas, leakage interference minimization can not be solved (even approximately) in polynomial time, unless P = NP.