Asynchronous Coordination Under Preferences And Constraints

Adaptive renaming can be viewed as a coordination task involving a set of asynchronous agents, each aiming at grabbing a single resource out of a set of resources Similarly, musical chairs is also defined as a coordination task involving a set of asynchronous agents, each aiming at picking one of a set of available resources, where every agent comes with an a priori preference for some resource. We foresee instances in which some combinations of resources are allowed, while others are disallowed.We model these constraints as an undirected graph whose nodes represent the resources, and an edge between two resources indicates that these two resources cannot be used simultaneously. In other words, the sets of resources that are allowed are those which form independent sets.We assume that each agent comes with an a priori preference for some resource. If an agent's preference is not in conflict with the preferences of the other agents, then this preference can be grabbed by the agent. Otherwise, the agents must coordinate to resolve their conflicts, and potentially choose non preferred resources. We investigate the following problem: given a graph, what is the maximum number of agents that can be accommodated subject to non-altruistic behaviors of early arriving agents?Just for cyclic constraints, the problem is surprisingly difficult. Indeed, we show that, intriguingly, the natural algorithm inspired from optimal solutions to adaptive renaming or musical chairs is sub-optimal for cycles, but proven to be at most 1 to the optimal. The main message of this paper is that finding optimal solutions to the coordination with constraints and preferences task requires to design "dynamic" algorithms, that is, algorithms of a completely different nature than the "static" algorithms used for, e.g., renaming.