Byzantine Convex Consensus: Preliminary Version

TL;DR

This paper introduces Byzantine convex consensus, allowing nodes in a distributed system to agree on a convex polytope within the convex hull of fault-free nodes' inputs, extending previous scalar and vector consensus algorithms.

Contribution

It generalizes approximate Byzantine consensus to convex polytopes in d-dimensional space, providing an asynchronous algorithm with optimal fault tolerance.

Findings

Provides a bound on the output convex polytope size.

Extends previous scalar and vector consensus algorithms.

Achieves asynchronous approximate consensus on convex polytopes.

Abstract

Much of the past work on asynchronous approximate Byzantine consensus has assumed scalar inputs at the nodes [3, 7]. Recent work has yielded approximate Byzantine consensus 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 vectors at the fault-free nodes [8, 12]. The d-dimensional vectors can be equivalently viewed as points in the d-dimensional Euclidean space. Thus, the algorithms in [8, 12] require the fault-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 polytope in the d-dimensional space, such that the decided polytope is within the convex hull of the input vectors at the fault-free nodes. We name this problem as Byzantine convex consensus (BCC), and present an asynchronous approximate BCC algorithm with optimal fault tolerance. Ideally, the goal here is to agree on a convex polytope that is as large as possible. While we do not claim that our algorithm satisfies this goal, we show a bound on the output convex polytope chosen by our algorithm.