Online interval scheduling: Randomized and multiprocessor cases

We consider the problem of scheduling a set of equal-length intervals arriving online, where each interval is associated with a weight and the objective is to maximize the total weight of completed intervals. An optimal 4-competitive algorithm has long been known in the deterministic case, but the randomized case remains open. We give the first randomized algorithm for this problem, achieving a competitive ratio of 3.618. We also prove a randomized lower bound of 4/3, which is an improvement over the previous 5/4 result, and a lower bound of 2 for a class of barely random algorithms which include our new algorithm. We also show that the techniques can be carried to the deterministic multiprocessor case, giving a 3.618-competitive 2-processor algorithm, a 5/4 lower bound for any number of processors, and a 2 lower bound for 2 processors.