Project - Team GANG Networks , Graphs and Algorithms
- Graph Decompositions
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 totally ordered by their desirability. 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, i.e., the restrictions on the ability to use some combinations of resources, 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 in the graph. E.g., renaming and musical chairs are specific cases where the graph is stable (i.e., it the empty graph containing no edges). As for musical chairs, 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? We entirely solve this problem under the restriction that agents which cannot grab their preferred resources must then choose a resource among the nodes of a predefined independent set. However, the general case, where agents which cannot grab their preferred resource are then free to choose any resource, is shown to be far more complex. In particular, 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. 1998 ACM Subject Classification D.1.3 Concurrent Programming (Distributed programming); F.2.2 Nonnumerical Algorithms and Problems.