Efficient, Optimal $k$-Leader Selection for Coherent, One-Dimensional Formations

TL;DR

This paper presents polynomial-time algorithms for optimal $k$-leader selection in one-dimensional path and ring graphs, improving scalability for consensus networks with noisy information.

Contribution

It introduces the first efficient, non-combinatorial methods to find optimal leader sets in 1D weighted graphs, specifically path and ring graphs.

Findings

Polynomial-time algorithms for path graphs ($O(n^3)$)

Polynomial-time algorithms for ring graphs ($O(kn^3)$)

Optimal leader selection reduces formation deviation effectively

Abstract

We study the problem of optimal leader selection in consensus networks with noisy relative information. The objective is to identify the set of $k$ leaders that minimizes the formation's deviation from the desired trajectory established by the leaders. 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.