Performance guarantees for greedy maximization of non-submodular controllability metrics

TL;DR

This paper establishes performance guarantees for greedy algorithms optimizing certain non-submodular controllability metrics in complex networks, using measures of submodularity ratio and curvature.

Contribution

It introduces a framework to provide performance bounds for greedy maximization of non-submodular functions related to network controllability metrics.

Findings

Performance guarantees are derived for greedy algorithms on non-submodular metrics.

The submodularity ratio and curvature are key to quantifying approximation quality.

Results extend the applicability of greedy methods beyond submodular functions.

Abstract

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.