Tractable approximation algorithms for the NP-hard problem of growing linear consensus networks

Abstract

In this paper, we investigate a network synthesis problem that aims to improve the performance of noisy linear consensus networks by establishing a few new interconnection links throughout the network. This network synthesis problem can be equivalently formulated as an optimization problem over a set of candidate graphs, which is inherently combinatorial and NP-hard. We propose an exact solution for this problem when only one new link needs to be established. In this case, a closed form expression is calculated for the optimal value of the systemic performance measure of the augmented network. When it is required to establish more than one interconnection link, we propose two tractable approximate methods with computable performance bounds. Moreover, in order to be able to calculate sub-optimality gaps of our proposed approximate algorithms, we quantify the best achievable performance bounds for the network synthesis problem.