Rigid Network Design Via Submodular Set Function Optimization

We consider the problem of constructing networks that exhibit desirable algebraic rigidity properties, which can provide significant performance improvements for associated formation shape control and localization tasks. We show that the network design problem can be formulated as a submodular set function optimization problem and propose greedy algorithms that achieve global optimality or an established near-optimality guarantee. We also consider the separate but related problem of selecting anchors for sensor network localization to optimize a metric of the error in the localization solutions. We show that an interesting metric is a modular set function, which allows a globally optimal selection to be obtained using a simple greedy algorithm. The results are illustrated via numerical examples, and we show that the methods scale to problems well beyond the capabilities of current state-of-the-art convex relaxation techniques.