Iterative Byzantine Vector Consensus in Incomplete Graphs

This work addresses

Byzantine vector consensus

, wherein the input at each process is a

d

-dimensional vector of reals, and each process is required to decide on a

decision vector

that is in the

convex hull

of the input vectors at the fault-free processes [9,12]. 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 [9,12] has addressed Byzantine vector consensus, and presented algorithms with optimal fault tolerance in complete graphs. This paper considers Byzantine vector consensus in incomplete graphs using

a restricted class

of iterative algorithms that maintain only a small amount of memory across iterations. For such algorithms, we 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.