Asynchronous Byzantine Approximate Consensus in Directed Networks

This paper considers the problem of approximate consensus in directed asynchronous message-passing networks where some nodes may become Byzantine faulty. We obtain a tight necessary and sufficient condition on the underlying directed communication network for asynchronous Byzantine approximate consensus to be achievable. Interestingly, this condition coincides with the tight condition for synchronous Byzantine exact consensus. Our consensus algorithm may be viewed as a non-trivial generalization of an algorithm previously proposed for the special case of complete networks. The tight condition and techniques identified in the paper shed light on the fundamental properties for solving approximate consensus in asynchronous directed networks.