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A set of stochastic matrices ${\cal P}$ is a consensus set if for everysequence of matrices $P(1), P(2), \ldots$ whose elements belong to ${\cal P}$and every initial state $x(0)$, the sequence of states defined by $x(t) = P(t)P(t-1) \cdots P(1) x(0)$ converges to a vector whose entries are all identical.In this paper, we introduce an "avoiding set condition" for compact sets ofmatrices and prove in our main theorem that this explicit combinatorialcondition is both necessary and sufficient for consensus. We show that severalof the conditions for consensus proposed in the literature can be directlyderived from the avoiding set condition. The avoiding set condition is easy tocheck with an elementary algorithm, and so our result also establishes thatconsensus is algorithmically decidable. Direct verification of the avoiding setcondition may require more than a polynomial time number of operations. This ishowever likely to be the case for any consensus checking algorithm since wealso prove in this paper that unless $P=NP$, consensus cannot be decided inpolynomial time.

How to decide consensus? A combinatorial necessary and sufficient condition and a proof that consensus is decidable but NP-hard