Dispersion, Capacitated Nodes, and the Power of a Trusted Shepherd

In this paper, we look at and expand the problems of dispersion and Byzantine dispersion of mobile robots on a graph, introduced by Augustine and Moses~Jr.~[ICDCN~2018] and by Molla, Mondal, and Moses~Jr.~[ALGOSENSORS~2020], respectively, to graphs where nodes have variable capacities. We use the idea of a single shepherd, a more powerful robot that will never act in a Byzantine manner, to achieve fast Byzantine dispersion, even when other robots may be strong Byzantine in nature. We also show the benefit of a shepherd for dispersion on capacitated graphs when no Byzantine robots are present.