Optimal k-leader selection for coherence and convergence rate in one-dimensional networks

Abstract

We study the problem of optimal leader selection in consensus networks under two performance measures: 1) formation coherence when subject to additive perturbations, as quantified by the steady-state variance of the deviation from the desired trajectory, and 2) convergence rate to a consensus value. The objective is to identify the set of $k$ leaders that optimizes the chosen performance measure. In both cases, an optimal leader set can be found by an exhaustive search over all possible leader sets; however, this approach is not scalable to large networks. In recent years, several works have proposed approximation algorithms to the $k$ -leader selection problem, yet the question of whether there exists an efficient, noncombinatorial method to identify the optimal leader set remains open. This work takes a first step toward answering this question. We show that, in 1-D weighted graphs, namely, path graphs and ring graphs, the $k$ -leader selection problem can be solved in polynomial time (in $k$ and the network size $n$ ). We give an $O(n^{3})$ solution for optimal $k$ -leader selection in path graphs and an $O(kn^{3})$ solution for optimal $k$ -leader selection in ring graphs.