Byzantine Convex Consensus: Preliminary Version

Much of the past work on asynchronous approximate Byzantine consensus has as-sumed scalar inputs at the nodes [3, 7]. 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 [8, 12]. The d-dimensional vectors can be equivalently viewed aspoints in the d-dimensional Euclidean space. Thus, the algorithms in [8, 12] require thefault-free nodes to decide on a point in the d-dimensional space. In this paper, we generalize the problem to allow the decision to be a convex polytopein the d-dimensional space, such that the decided polytope is within the convex hull ofthe input vectors at the fault-free nodes. We name this problem as Byzantine convexconsensus (BCC), and present an asynchronous approximate BCC algorithm with op-timal fault tolerance. Ideally, the goal here is to agree on a convex polytope that is aslarge as possible. While we do not claim that our algorithm satisfies this goal, we showa bound on the output convex polytope chosen by our algorithm. ∗We present an optimal algorithm in our follow-up work [11].†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.