Abstract
We study the online preemptive scheduling of intervals and jobs (with restarts). Each interval or job has an arrival time, a deadline, a length and a weight. The objective is to maximize the total weight of completed intervals or jobs. While the deterministic case for intervals was settled a long time ago, the randomized case remains open. In this paper we first give a 2-competitive randomized algorithm for the case of equal length intervals. The algorithm is barely random in the sense that it randomly chooses between two deterministic algorithms at the beginning and then sticks with it thereafter. Then we extend the algorithm to cover several other cases of interval scheduling including monotone instances, C-benevolent instances and D-benevolent instances, giving the same competitive ratio. These algorithms are surprisingly simple but have the best competitive ratio against all previous (fully or barely) randomized algorithms. Next we extend the idea to give a 3-competitive algorithm for equal length jobs. Finally, we prove a lower bound of 2 on the competitive ratio of all barely random algorithms that choose between two deterministic algorithms for scheduling equal length intervals (and hence jobs). A preliminary version of this paper appeared in Fung et al. (The 6th International Workshop on Approximation and Online Algorithmspp, vol. 5426, pp. 53–66, 2008).
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