  In this paper, we show that the optimal receive antenna subset selection
problem for maximizing the mutual information in a point-to-point MIMO system
is sub-modular. Consequently, a greedy step-wise optimization approach, where
at each step an antenna that maximizes the incremental gain is added to the
existing antenna subset, is guaranteed to be within a (1 - 1/e) fraction of the
global optimal value. For a single antenna equipped source and destination with
multiple relays, we show that the relay antenna selection problem to maximize
the mutual information is modular, when complete channel state information is
available at the relays. As a result a greedy step-wise optimization approach
leads to an optimal solution for the relay antenna selection problem with
linear complexity in comparison to the brute force search that incurs
exponential complexity.
