We study the fundamental scheduling problem $1\|\sum p_jU_j$. Given a set of $n$ jobs with processing times $p_j$ and deadlines $d_j$, the problem is to select a subset of jobs such that the total processing time is maximized without violating the deadlines. In the midst of a flourishing line of research, Fischer and Wennmann have recently devised the sought-after $\widetilde O(P)$-time algorithm, where $P = \sum p_j$ is the total processing time of all jobs. This running time is optimal as it matches conditional lower bounds based on popular conjectures. However, $P$ is not the sole parameter one could parameterize the running time by. Indeed, they explicitly leave open the question of whether a running time of $\widetilde O(n + \max d_j)$ or even $\widetilde O(n + \max p_j)$ is possible. In this work, we show, somewhat surprisingly, that by a refined implementation of their original algorithm, one can obtain the asked-for $\widetilde O(n + \max d_j)$-time algorithm.

翻译：我们研究基本调度问题 $1\|\sum p_jU_j$。给定一组具有处理时间 $p_j$ 和截止日期 $d_j$ 的 $n$ 个作业，该问题是在不违反截止日期的前提下最大化总处理时间。在活跃的研究脉络中，Fischer 和 Wennmann 最近设计了备受期待的 $\widetilde O(P)$ 时间算法，其中 $P = \sum p_j$ 是所有作业的总处理时间。该运行时间是最优的，因为它与基于流行推测的条件下界相匹配。然而，$P$ 并非唯一可用于参数化运行时间的参数。事实上，他们明确留下了关于 $\widetilde O(n + \max d_j)$ 甚至 $\widetilde O(n + \max p_j)$ 运行时间是否可能的开放问题。在本工作中，我们证明，令人意外的是，通过对其原始算法进行精细实现，可以获得所要求的 $\widetilde O(n + \max d_j)$ 时间算法。