Optimal k-Leader Selection for Coherence and Convergence Rate in One-Dimensional Networks

TL;DR

This paper investigates the optimal selection of leader nodes in one-dimensional consensus networks to improve coherence and convergence, providing polynomial-time algorithms for path and ring graphs.

Contribution

It introduces efficient polynomial-time algorithms for optimal k-leader selection in one-dimensional networks, solving a problem previously approached only by exhaustive search.

Findings

Polynomial-time algorithms for path graphs

Polynomial-time algorithms for ring graphs

Improved scalability for leader selection in 1D 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, non-combinatorial method to identify the optimal leader set remains open. This work takes a first step towards answering this question. We show that, in one-dimensional weighted graphs, namely path graphs and ring graphs, the $k$-leader selection problem can be solved in polynomial time (in both $k$ 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.