Greed is good: Leveraging submodularity for antenna selection in Massive MIMO

We consider the NP-Hard problem of performing antenna selection in the downlink of a single cell, multi-user Massive MIMO system by maximizing the downlink channel capacity with a specified power allocation matrix subject to a cardinality constraint on the number of selected antennas. Prior work has focused on using convex relaxation coupled with fractional rounding in an attempt to obtain high quality sub-optimal solutions for this problem in polynomial-time. However, one cannot quantify the sub-optimality of the solution obtained via this approach for an arbitrary problem instance, which, in addition, is also computationally demanding to determine for a large-scale antenna system. In this paper, we show that the objective function of the antenna selection problem is monotone submodular, which implies that a simple greedy algorithm can be used to guarantee a constant (1-1/e)-factor approximation for all problem instances. The merits of using this approach are illustrated via simulations where the greedy algorithm returns high quality solutions in all cases at significantly lower complexity relative to convex relaxation based approaches.