We consider the Consensus problem on arbitrary oblivious message adversaries. A message adversary models a communication network whose topology evolves from round to round. We make no assumptions about the possible topologies. A message adversary is oblivious if the set of possible topologies is the same at every round.
We give the first complete necessary and sufficient condition for message adversaries that admits a Consensus algorithm. For the necessary part we present a specialized bivalency proof. For the sufficiency part, we present a new algorithm that is based upon reconstructing a partial, but significant, view of the actual communications that occurred during the evolution of the network. This reconstruction might be of independent interest.
