  Approximate byzantine consensus is a fundamental problem of distributed
computing. This paper presents a novel algorithm for approximate byzantine
consensus, called Relay-ABC. The algorithm allows machines to achieve
approximate consensus to arbitrary exactness in the presence of byzantine
failures. The algorithm relies on the usage of a relayed messaging system and
signed messages with unforgeable signatures that are unique to each node. The
use of signatures and relays allows the strict necessary network conditions of
traditional approximate byzantine consensus algorithms to be circumvented.
  We also provide theoretical guarantees of validity and convergence for
Relay-ABC. To do this, we utilize the idea that the iteration of states in the
network can be modeled by a sequence of transition matrices. We extend previous
methods, which use transition matrices to prove ABC convergence, by having each
state vector model not just one iteration, but a set of $D$ iterations, where
$D$ is a diameter property of the graph. This allows us to accurately model the
delays of messages inherent within the relay system.
