Iterative Byzantine Vector Consensus in Incomplete Graphs

TL;DR

This paper investigates iterative Byzantine vector consensus in incomplete graphs, establishing conditions for solvability, and proposes an algorithm that is correct under these conditions, advancing understanding of fault-tolerant distributed consensus.

Contribution

It introduces necessary and sufficient conditions for iterative Byzantine vector consensus in incomplete graphs and presents a new algorithm proven correct under these conditions.

Findings

Necessary and sufficient conditions for consensus in incomplete graphs.

Proposed an iterative consensus algorithm with correctness proof.

Identified gaps between necessary and sufficient conditions for d > 1.

Abstract

This work addresses Byzantine vector consensus (BVC), wherein the input at each process is a d-dimensional vector of reals, and each process is expected to decide on a decision vector that is in the convex hull of the input vectors at the fault-free processes [3, 8]. The input vector at each process may also be viewed as a point in the d-dimensional Euclidean space R^d, where d > 0 is a finite integer. Recent work [3, 8] has addressed Byzantine vector consensus in systems that can be modeled by a complete graph. This paper considers Byzantine vector consensus in incomplete graphs. In particular, we address a particular class of iterative algorithms in incomplete graphs, and prove a necessary condition, and a sufficient condition, for the graphs to be able to solve the vector consensus problem iteratively. We present an iterative Byzantine vector consensus algorithm, and prove it correct under the sufficient condition. The necessary condition presented in this paper for vector consensus does not match with the sufficient condition for d > 1; thus, a weaker condition may potentially suffice for Byzantine vector consensus.