Gracefully Degrading Consensus and k-Set Agreement in Directed Dynamic Networks

We study distributed agreement in synchronous directed dynamic networks, where an omniscient message adversary controls the presence/absence of communication links. We prove that consensus is impossible under a message adversary that guarantees weak connectivity only, and introduce vertex-stable root components (VSRCs) as a means for circumventing this impossibility: A VSRC(k, d) message adversary guarantees that, eventually, there is an interval of d consecutive rounds where every communication graph contains at most k strongly connected components consisting of the same processes (with possibly varying interconnect topology), which have at most out-going links to the remaining processes. We present a consensus algorithm that works correctly under a VSRC(1, 4H + 2) message adversary, where H is the dynamic causal network diameter. Our algorithm maintains local estimates of the communication graphs, and applies techniques for detecting network stability and univalent system configurations. Several related impossibility results and lower bounds, in particular, that neither a VSRC(1,H − 1) message adversary nor a VSRC(2,∞) one allow to solve consensus, reveal that there is not much hope to deal with (much) stronger message adversaries here. However, we show that gracefully degrading consensus, which degrades to general k-set agreement in case of unfavorable network conditions, allows to cope with stronger message adversaries: We provide a k-uniform k-set agreement algorithm, where the number of system-wide decision values k is not encoded in the algorithm, but rather determined by the actual power of the message adversary in a run: Our algorithm guarantees at most k decision values under a VSRC(n, d) + MAJINF(k) message adversary, which combines VSRC(n, d) (with some small value of d, ensuring termination) with some information flow guarantee MAJINF(k) between certain VSRCs (ensuring k-agreement). Since related impossibility results reveal that a VSRC(k, d) message adversary is too strong for solving k-set agreement and that some information flow between VSRCs is mandatory for this purpose as well, our results provide a significant step towards the exact solvability/impossibility border of general k-set agreement in directed dynamic networks.