具有区间预测和昂贵抢占的单机调度
Single-Machine Scheduling with Interval Predictions and Costly Preemption
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中文总结 AI 辅助
针对加工时间未知且带报告区间的单机调度问题，提出几何回退策略，推导最坏情况系数并给出最优深度选择，为区间预测提供性能保证。
中文摘要 AI 辅助
我们研究了当加工时间未知但每个作业都带有一个报告区间时，单机总完工时间问题 $1||\sum C_j$。作业最初按照报告上界的非递减顺序进行加工。如果一个作业在接受了那么多服务后仍未完成，则说明报告上界已被违反，策略将切换为可恢复的几何回退。每次中断一个未完成的作业都会产生一个加性惩罚 $\kappa$。对于剩余 $m$ 个作业的集合，已知大小比率 $D=s_1/s_0$，归一化中断惩罚 $\lambda=\kappa/s_0$，以及几何深度 $K$，我们推导出一个显式的最坏情况系数 $\Psi_{m,D,\lambda}(K)$。当 $\lambda>0$ 时，存在一个有限的最优深度，并且可以在触发后根据观察到的未完成作业数量来选择；最优深度随中断惩罚的增加而减小，随剩余集合大小的增加而增大。我们还推导了当所有加工时间都位于已知有界范围内时，任意非抢占列表的精确最坏情况系数 $R_{n,D}$。这些界为有效区间、单区间失败、任意报告、无触发结果和随机实例提供了保证。在单失败机制下，它们还给出了一个精确条件，在该条件下，已证明的回退保证小于有界范围继续保证。
英文摘要
We study the single-machine total-completion-time problem $1||\sum C_j$ when processing times are unknown but each job comes with a reported interval. Jobs are initially processed in nondecreasing order of reported upper bound. If a job is still unfinished after receiving that much service, the reported upper bound has been violated and the policy switches to a resumable geometric fallback. Each interruption of an unfinished job incurs an additive penalty $κ$. For a residual set of $m$ jobs, known size ratio $D=s_1/s_0$, normalized interruption penalty $λ=κ/s_0$, and geometric depth $K$, we derive an explicit worst-case coefficient $Ψ_{m,D,λ}(K)$. When $λ>0$, a finite minimizing depth exists and can be chosen after the trigger from the observed number of unfinished jobs; the optimal depth decreases with the interruption penalty and increases with the residual set size. We also derive the exact worst-case coefficient $R_{n,D}$ for an arbitrary nonpreemptive list when all processing times lie in a known bounded range. These bounds give guarantees for valid intervals, a single interval failure, arbitrary reports, no-trigger outcomes, and random instances. In the single-failure regime, they also give a precise condition under which the proved fallback guarantee is smaller than the bounded-range continuation guarantee.
