We consider the problem of approximate consensus in mobile networks containing Byzantine nodes. We assume that each correct node can communicate only with its neighbors and has no knowledge of the global topology. As all nodes have moving ability, the topology is dynamic. The number of Byzantine nodes is bounded by f and known by all correct nodes. We first introduce an approximate Byzantine consensus protocol which is based on the linear iteration method. As nodes are allowed to collect information during several consecutive rounds, moving gives them the opportunity to gather more values. We propose a novel sufficient and necessary condition to guarantee the final convergence of the consensus protocol. The requirement expressed by our condition is not "universal": in each phase it affects only a single correct node. More precisely, at least one correct node among those that propose either the minimum or the maximum value which is present in the network, has to receive enough messages (quantity constraint) with either higher or lower values (quality constraint). Of course, nodes' motion should not prevent this requirement to be fulfilled. Our conclusion shows that the proposed condition can be satisfied if the total number of nodes is greater than 3f+1.
This paper investigates the task solvability of mobile robot systems subject to Byzantine faults. We first consider the gathering problem, which requires all robots to meet in finite time at a non-predefined location. It is known that the solvability of Byzantine gathering strongly depends on a number of system attributes, such as synchrony, the number of Byzantine robots, scheduling strategy, obliviousness, orientation of local coordinate systems and so on. However, the complete characterization of the attributes making Byzantine gathering solvable still remains open. In this paper, we show strong impossibility results of Byzantine gathering. Namely, we prove that Byzantine gathering is impossible even if we assume one Byzantine fault, an atomic execution system, the n-bounded centralized scheduler, non-oblivious robots, instantaneous movements and a common orientation of local coordinate systems (where n denote the number of correct robots). Those hypotheses are much weaker than used in previous work, inducing a much stronger impossibility result. At the core of our impossibility result is a reduction from the distributed consensus problem in asynchronous shared-memory systems. In more details, we newly construct a generic reduction scheme based on the distributed BG-simulation. Interestingly, because of its versatility, we can easily extend our impossibility result for general pattern formation problems.