Tight Bounds for Connectivity and Set Agreement in Byzantine Synchronous Systems

In this paper, we show that the protocol complex of a Byzantine synchronous system can remain (k-1)-connected for up to ceil(t/k) rounds, where t is the maximum number of Byzantine processes, and t >= k >= 1. This topological property implies that ceil(t/k) + 1 rounds are necessary to solve k-set agreement in Byzantine synchronous systems, compared to floor(t/k) + 1 rounds in synchronous crash-failure systems. We also show that our connectivity bound is tight as we indicate solutions to Byzantine k-set agreement in exactly ceil(t/k) + 1 synchronous rounds, at least when n is suitably large compared to t. In conclusion, we see how Byzantine failures can potentially require one extra round to solve k-set agreement, and, for n suitably large compared to t, at most that.