Performance guarantees for greedy maximization of non-submodular controllability metrics

A key problem in emerging complex cyber-physical networks is the design of information and control topologies, including sensor and actuator selection and communication network design. These problems can be posed as combinatorial set function optimization problems to maximize a dynamic performance metric for the network. Some systems and control metrics feature a property called submodularity, which allows simple greedy algorithms to obtain provably near-optimal topology designs. However, many important metrics lack submodularity and therefore lack provable guarantees for using a greedy optimization approach. Here we show that performance guarantees can be obtained for greedy maximization of certain non-submodular functions of the controllability and observability Gramians. Our results are based on two key quantities: the submodularity ratio, which quantifies how far a set function is from being submodular, and the curvature, which quantifies how far a set function is from being supermodular.