Deterministic Local Algorithms, Unique Identifiers, and Fractional Graph Colouring

In the fractional graph colouring problem, the task is to schedule the activities of the nodes so that each node is active for 1 time unit in total, and at each point of time the set of active nodes forms an independent set. We show that for any α > 1 there exists a deterministic distributed algorithm that finds a fractional graph colouring of length at most α(∆ + 1) in any graph in one synchronous communication round; here ∆ is the maximum degree of the graph. The result is near-tight, as there are graphs in which the optimal solution has length ∆ + 1. The result is, of course, too good to be true. The usual definitions of scheduling problems (fractional graph colouring, fractional domatic partition, etc.) in a distributed setting leave a loophole that can be exploited in the design of distributed algorithms: the size of the local output is not bounded. Our algorithm produces an output that seems to be perfectly good by the usual standards but it is impractical, as the schedule of each node consists of a very large number of short periods of activity. More generally, the algorithm demonstrates that when we study distributed algorithms for scheduling problems, we can choose virtually any trade-off between the following three parameters: T , the running time of the algorithm, `, the length of the schedule, and κ, the maximum number of periods of activity for any single node. Here ` is the objective function of the optimisation problem, while κ captures the “subjective” quality of the solution. If we study, for example, bounded-degree graphs, we can trivially keep T and κ constant, at the cost of a large `, or we can keep κ and ` constant, at the cost of a large T . Our algorithm shows that yet another trade-off is possible: we can keep T and ` constant at the cost of a large κ.