AI 中文总结
针对单台机器的在线繁忙时间调度问题，提出竞争比小于3.72的随机化在线算法，证明随机化算法竞争比下界为e，同时给出作业长度统一时确定性算法的竞争比下界。
AI 中文摘要
我们研究具有非抢占式作业的单台无限容量机器上的在线繁忙时间调度模型。在我们的设定中，柔性作业在线到达，其处理时间和截止日期在作业到达时才被算法知晓。目标是在机器上调度作业以在截止日期前完成所有作业，同时最小化机器开启的总时间（繁忙时间，在此设定中也称为跨度）。我们提出了一种随机化在线算法，针对无知敌手的竞争比为1 + e < 3.72，其中e为欧拉数，并且证明了针对无知敌手，任何随机化算法的竞争比都无法优于e，即使是允许重启作业的算法也如此。以往的工作要么处理特殊情况，要么处理确定性算法，已知确定性算法的竞争比上界为5，下界为4。我们的发现为在线能耗感知调度中的随机化提供了新的见解。在作业长度统一且算法启动的作业必须完成的设定中，我们证明即使作业是合意的，任何确定性算法的性能也无法优于2。
英文摘要
We study the online Busy Time scheduling model on a single machine of unbounded capacity, with non-preemptive jobs. In our setting, flexible jobs arrive online with a processing time and deadline, both of which become known to the algorithm at the job's arrival time. The goal is to schedule jobs on the machine to finish all jobs by their deadlines, so that the total time when the machine is turned on (busy time, also called span in this setting) is minimized. We present a randomized online algorithm with an exact competitive ratio of exactly (1 + e) < 3.72, where e is Euler's number, against an oblivious adversary, and show that no randomized algorithm can have a competitive ratio better than $e$ against an oblivious adversary. This lower bound holds even for algorithms that are allowed to restart jobs and are given lookahead. Previous work either dealt with special cases, or with deterministic algorithms, for which the known upper bound on the competitive ratio is 5 and the known lower bound is 4. Our findings offer fresh insights into randomization in online energy-aware scheduling. In the setting where jobs have uniform processing times, and a job that is started by the algorithm must be finished, we show that no deterministic algorithm can do better than 2, even when the jobs are agreeable. This deterministic lower bound also holds for uniform processing times and restarts in the scenario where a job's deadline is only revealed at its starting deadline. For Capacitated Busy Time with p_max-lookahead, we obtain a deterministic online algorithm with competitive ratio at most 8 and a randomized online algorithm with competitive ratio at most (4+ e) < 6.72, against an oblivious adversary.
