This paper proves a necessary and sufficient condition for the existence of iterative algorithms that achieve approximate Byzantine consensus in arbitrary directed graphs, where each directed edge represents a communication channel between a pair of nodes. The class of iterative algorithms considered in this paper ensures that, after each iteration of the algorithm, the state of each fault-free node remains in the convex hull of the states of the fault-free nodes at the end of the previous iteration. The following convergence requirement is imposed: for any ∈ ＞ 0, after a sufficiently large number of iterations, the states of the fault-free nodes are guaranteed to be within ∈ of each other. To the best of our knowledge, tight necessary and sufficient conditions for the existence of such iterative consensus algorithms in synchronous arbitrary point-to-point networks in presence of Byzantine faults have not been developed previously. The methodology and results presented in this paper can also be extended to asynchronous systems.
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