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(n3) solution for
optimal k-leader selection in path graphs and an O(kn3) solution for
optimal k-leader selection in ring graphs.Comment: 7 pages, 5 figures, submitted to ECC1