On the cost of deciding consensus

We study the computational complexity of a general consensus problem for switched systems. A set of n × n stochastic matrices {P₁, ..., P_k} is a consensus set if for every switching map τ : N → {1, ..., k} and for every initial state x(0), the sequence of states defined by x(t + 1) = P_τ(t)x(t) converges to a state whose entries are all identical. We show in this paper that, unless P = NP, the problem of determining if a set of matrices is a consensus set cannot be decided in polynomial-time. As a consequence, unless P = NP, it is not possible to give efficiently checkable necessary and sufficient conditions for consensus. This provides a possible explanation for the absence of such conditions in the current literature on consensus. On the positive side, we provide a simple algorithm which checks whether {P₁, ..., P_k} is a consensus set in a number of operations which scales as a doubly exponential in n.