  In this work, we study the approximate consensus problem in asynchronous
message-passing networks where some nodes may become Byzantine faulty. We
answer an open problem raised by Tseng and Vaidya, 2012, proposing the first
algorithm of optimal resilience for directed networks. Interestingly, our
results show that the tight condition on the underlying communication networks
for asynchronous Byzantine approximate consensus coincides with the tight
condition for synchronous Byzantine exact consensus. Our results can be viewed
as a non-trivial generalization of the algorithm by Abraham et al., 2004, which
applies to 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.
