Reaching Approximate Byzantine Consensus with Multi-hop Communication

Abstract

We address the problem of reaching approximate consensus in the presence of Byzantine faults in a synchronous system. We analyze iterative algorithms that maintain minimal state, and impose the constraint that in each iteration the nodes may only communicate with other nodes that are up to l hops away. For a given l, we prove a necessary and sufficient condition on the network structure for the existence of correct iterative algorithms that achieve approximate Byzantine consensus. We prove sufficiency of the condition by designing a correct algorithm, which uses a trim function based on a minimal messages cover property introduced in this paper. Our necessary and sufficient condition generalizes the tight condition identified in prior work for $l=1$. For $l\ge l^*$, where $l^*$ is the length of a longest cycle-free path in the given network, our condition is equivalent to the necessary and sufficient conditions for exact consensus in undirected and directed networks both.

This research is supported in part by National Science Foundation awards NSF 1329681 and 1421918. Any opinions, findings, and conclusions or recommendations expressed here are those of the authors and do not necessarily reflect the views of the funding agencies or the U.S. government.