Interference forms a major challenge in our understanding of the capacity of wireless networks and our ability to achieve this capacity.... Show moreInterference forms a major challenge in our understanding of the capacity of wireless networks and our ability to achieve this capacity. Rather than scheduling transmissions to avoid interference, recent techniques allow for interference to be neutralized and for simultaneous transmission of messages.Linear interference alignment in MIMO networks is the technique of aligning messages, by the transmitters through the use of precoding matrices, so that the undesired messages occupy some minimal sub-space upon their arrival at an unintended receiver. The overlapping of the sub-spaces where these interfering messages fall allows the receiver to neutralize them with minimal dedication of its resources through the application of a decoding matrix.The linear interference alignment problem is to design these precoding and decoding matrices. It has been shown to be NP-hard in the literature.A network is called feasible if such a solution exists. Even deciding whether some network instance is feasible, is non-trivial. The problem of deciding feasibility was shown to be NP-hard in the literature, for constant channel coefficients.We focus on finding efficient and robust feasibility tests in the case of generic channels, where the computational complexity is unknown. We provide efficient and robust tests for the necessary condition of properness, which had previously been identified in the literature but given no efficient tests in the general case.We identify several conditions, each being sufficient for feasibility. We study their relationships and the computational complexity of testing for them. We provide polynomial-time maximum flow test for one sufficient condition in the case of uniform demands. In the case of uniform demands which divide the number of antennas at all receivers or all transmitters, we show that these sufficient and necessary conditions are equivalent with feasibility, thereby admitting efficient maximum-flow tests.We identify a subset of feasible instances where the decoding and precoding matrices can be designed in polynomial-time. Furthermore, we show that any proper instance is within a constant factor of a one of these instances. Then, we provide efficient constant approximation algorithms for the problems of maximizing demand and minimizing antennas such that an instance is feasible. Show less
