Optimally handling commitment issues in online throughput maximization
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We consider a fundamental online scheduling problem in which jobs with processing times and deadlines arrive online over time at their release dates. The task is to determine a feasible preemptive schedule on a single machine that maximizes the number of jobs that complete before their deadline. Due to strong impossibility results for competitive analysis, it is commonly required that jobs contain some slack ε>0, which means that the feasible time window for scheduling a job is at least 1+ε times its processing time. In this paper, we resolve the question on how to handle commitment requirements which enforce that a scheduler has to guarantee at a certain point in time the completion of admitted jobs. This is very relevant, e.g., in providing cloud-computing services and disallows last-minute rejections of critical tasks. We give an algorithm with an optimal competitive ratio of Θ(1/ε) for the online throughput maximization problem when a scheduler must commit upon starting a job. Somewhat surprisingly, this is the same optimal performance bound (up to constants) as for scheduling without commitment. If commitment decisions must be made before a job's slack becomes less than a δ-fraction of its size, we prove a competitive ratio of O(ε/((ε - δ)δ)) for 0 < δ < ε. This result interpolates between commitment upon starting a job and commitment upon arrival. For the latter commitment model, it is known that no (randomized) online algorithms does admit any bounded competitive ratio.
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