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. The NP-hardness of this problem had been left open by Blondel and Olshevsky [V. D. Blondel and A. Olshevsky, SIAM J. Control Optim., 52 (2014), pp. 2707-2726].