Byzantine Convex Consensus: An Optimal Algorithm

Much of the past work on asynchronous approximate Byzantine consensus has as-sumed scalar inputs at the nodes [4, 8]. Recent work has yielded approximate Byzantineconsensus algorithms for the case when the input at each node is a d-dimensional vector,and the nodes must reach consensus on a vector in the convex hull of the input vectorsat the fault-free nodes [9, 13]. The d-dimensional vectors can be equivalently viewed aspoints in the d-dimensional Euclidean space. Thus, the algorithms in [9, 13] require thefault-free nodes to decide on a point in the d-dimensional space. In our recent work [12], we proposed a generalization of the consensus problem,namely Byzantine convex consensus (BCC), which allows the decision to be a convexpolytope in the d-dimensional space, such that the decided polytope is within the convexhull of the input vectors at the fault-free nodes. We also presented an asynchronousapproximate BCC algorithm. In this paper, we propose a new BCC algorithm with optimal fault-tolerance thatalso agrees on a convex polytope that is as large as possible under adversarial conditions.Our prior work [12] does not guarantee the optimality of the output polytope. ∗This research is supported in part by National Science Foundation award CNS 1059540. Anyopinions, findings, and conclusions or recommendations expressed here are those of the authors anddo not necessarily reflect the views of the funding agencies or the U.S. government.