How to Decide Consensus? A Combinatorial Necessary and Sufficient Condition and a Proof that Consensus is Decidable but NP-Hard

@article{Blondel2014HowTD,
title={How to Decide Consensus? A Combinatorial Necessary and Sufficient Condition and a Proof that Consensus is Decidable but NP-Hard},
author={Vincent D. Blondel and Alexander Olshevsky},
journal={SIAM J. Control and Optimization},
year={2014},
volume={52},
pages={2707-2726}
}

A set of stochastic matrices P is a consensus set if for every sequence of matrices P (1), P (2), . . . whose elements belong to P and every initial state x(0), the sequence of states defined by x(t) = P (t)P (t − 1) · · ·P (1)x(0) converges to a vector whose entries are all identical. In this paper, we introduce an “avoiding set condition” for compact sets of matrices and prove in our main theorem that this explicit combinatorial condition is both necessary and sufficient for consensus. We… CONTINUE READING