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：我们研究基础调度问题 $1\|\sum p_jU_j$。给定一组具有处理时间 $p_j$ 和截止期限 $d_j$ 的 $n$ 个作业，该问题要求在不违反截止期限的前提下，最大化所选择作业的总处理时间。在这一蓬勃发展的研究脉络中，Fischer 与 Wennmann 近期设计出了广受期待的 $\widetilde O(P)$ 时间算法，其中 $P = \sum p_j$ 为所有作业的总处理时间。该运行时间已达到最优，因为它与基于流行猜想的条件下界相匹配。然而，$P$ 并非可参数化运行时间的唯一参量。事实上，他们明确提出了一个开放性问题：能否实现 $\widetilde O(n + \max d_j)$ 甚至 $\widetilde O(n + \max p_j)$ 的运行时间。在本工作中，我们出人意料地表明，通过对其原始算法进行精细化实现，即可获得所要求的 $\widetilde O(n + \max d_j)$ 时间算法。