Exact Byzantine consensus problem requires that non-faulty processes reach agreement on a decision (or output) that is in the convex hull of the inputs at the non-faulty processes. It is well-known that exact consensus is impossible in an asynchronous system in presence of faults, and in a synchronous system, n>=3f+1 is tight on the number of processes to achieve exact Byzantine consensus with scalar inputs, in presence of up to f Byzantine faulty processes. Recent work has shown that when the inputs are d-dimensional vectors of reals, n>=max(3f+1,(d+1)f+1) is tight to achieve exact Byzantine consensus in synchronous systems, and n>= (d+2)f+1 for approximate Byzantine consensus in asynchronous systems. Due to the dependence of the lower bound on vector dimension d, the number of processes necessary becomes large when the vector dimension is large. With the hope of reducing the lower bound on n, we consider two relaxed versions of Byzantine vector consensus: k-Relaxed Byzantine vector consensus and (delta,p)-Relaxed Byzantine vector consensus. In k-relaxed consensus, the validity condition requires that the output must be in the convex hull of projection of the inputs onto any subset of k-dimensions of the vectors. For (delta,p)-consensus the validity condition requires that the output must be within distance delta of the convex hull of the inputs of the non-faulty processes, where L_p norm is used as the distance metric. For (delta,p)-consensus, we consider two versions: in one version, delta is a constant, and in the second version, delta is a function of the inputs themselves. We show that for k-relaxed consensus and (delta,p)-consensus with constant delta>=0, the bound on n is identical to the bound stated above for the original vector consensus problem. On the other hand, when delta depends on the inputs, we show that the bound on n is smaller when d>=3.
The Byzantine vector consensus (BVC) problem (also called multidimensional consensus) was introduced by Mendes and Herlihy @cite_19 and Vaidya and Garg @cite_2 . Tight bounds on number of processes @math for Byzantine vector consensus have been obtained for synchronous @cite_2 and asynchronous @cite_19 @cite_2 systems both, when the network is a complete graph. A necessary condition and a sufficient condition for iterative byzantine vector consensus were derived by Vaidya @cite_4 , however, there is a gap between these necessary and sufficient conditions.
@cite_19 @cite_4 @cite_2
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The problem of e-approximate agreement in Byzantine asynchronous systems is well-understood when all values lie on the real line. In this paper, we generalize the problem to consider values that lie in Rm, for m ≥ 1, and present an optimal protocol in regard to fault tolerance. Our scenario is the following. Processes start with values in Rm, for m ≥ 1, and communicate via message-passing. The system is asynchronous: there is no upper bound on processes' relative speeds or on message delay. Some faulty processes can display arbitrarily malicious (i.e. Byzantine) behavior. Non-faulty processes must decide on values that are: (1) in Rm; (2) within distance e of each other; and (3) in the convex hull of the non-faulty processes' inputs. We give an algorithm with a matching lower bound on fault tolerance: we require n > t(m+2), where n is the number of processes, t is the number of Byzantine processes, and input and output values reside in Rm. Non-faulty processes send O(n2 d log(m e max δ(d): 1 ≤ d ≤ m )) messages in total, where δ(d) is the range of non-faulty inputs projected at coordinate d. The Byzantine processes do not affect the algorithm's running time. 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. Consider a network of n processes each of which has a d-dimensional vector of reals as its input. Each process can communicate directly with all the processes in the system; thus the communication network is a complete graph. All the communication channels are reliable and FIFO (first-in-first-out). The problem of Byzantine vector consensus (BVC) requires agreement on a d-dimensional vector that is in the convex hull of the d-dimensional input vectors at the non-faulty processes. We obtain the following results for Byzantine vector consensus in complete graphs while tolerating up to f Byzantine failures: * We prove that in a synchronous system, n >= max(3f+1, (d+1)f+1) is necessary and sufficient for achieving Byzantine vector consensus. * In an asynchronous system, it is known that exact consensus is impossible in presence of faulty processes. For an asynchronous system, we prove that n >= (d+2)f+1 is necessary and sufficient to achieve approximate Byzantine vector consensus. Our sufficiency proofs are constructive. We show sufficiency by providing explicit algorithms that solve exact BVC in synchronous systems, and approximate BVC in asynchronous systems. We also obtain tight bounds on the number of processes for achieving BVC using algorithms that are restricted to a simpler communication pattern.