Efficient Algorithms for the Consensus Decision Problem

TL;DR

This paper develops efficient algorithms to determine the convergence of switched consensus systems under various switching conditions, providing new polynomial-time solutions for specific cases previously thought to be NP-hard.

Contribution

It introduces necessary and sufficient conditions for convergence problems and proves polynomial-time solvability for systems switching between two undirected subsystems.

Findings

Provided singly exponential time algorithms for convergence verification.

Established polynomial-time solvability for two-graph switching systems.

Challenged previous assumptions of NP-hardness for certain consensus problems.

Abstract

We address the problem of determining if a discrete time switched consensus system converges for any switching sequence and that of determining if it converges for at least one switching sequence. For these two problems, we provide necessary and sufficient conditions that can be checked in singly exponential time. As a side result, we prove the existence of a polynomial time algorithm for the first problem when the system switches between only two subsystems whose corresponding graphs are undirected, a problem that had been suggested to be NP-hard by Blondel and Olshevsky.