Byzantine Convex Consensus: An Optimal Algorithm

TL;DR

This paper introduces an optimal Byzantine convex consensus algorithm that guarantees fault-tolerance and maximizes the decision polytope size in asynchronous networks, improving upon previous non-optimal solutions.

Contribution

It presents a new Byzantine convex consensus algorithm with optimal fault-tolerance that ensures the largest possible convex polytope as output, advancing prior work which lacked this optimality.

Findings

Achieves optimal fault-tolerance in asynchronous Byzantine convex consensus

Guarantees the output polytope is as large as possible under adversarial conditions

Improves upon previous algorithms by ensuring optimality of the consensus polytope

Abstract

Much of the past work on asynchronous approximate Byzantine consensus has assumed scalar inputs at the nodes [4, 8]. 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 [9, 13]. The d-dimensional vectors can be equivalently viewed as points in the d-dimensional Euclidean space. Thus, the algorithms in [9, 13] require the fault-free nodes to decide on a point in the d-dimensional space. In our recent work [arXiv:/1307.1051], we proposed a generalization of the consensus problem, namely Byzantine convex consensus (BCC), which allows 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 also presented an asynchronous approximate BCC algorithm. In this paper, we propose a new BCC algorithm with optimal fault-tolerance that also agrees on a convex polytope that is as large as possible under adversarial conditions. Our prior work [arXiv:/1307.1051] does not guarantee the optimality of the output polytope.