The Art of Staying Ahead of Deadlines: Improved Algorithms for the Minimum Tardy Processing Time

We study the fundamental scheduling problem $1|\sum pjUj$. Given a set of $n$ jobs with processing times $pj$ and deadlines $dj$, 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 $P = \sum pj$ 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 $\widetilde O(n + \max dj)$ or even $\widetilde O(n + \max pj)$ is possible. In this work, we show, somewhat surprisingly, that by a refined implementation of their original algorithm, one can obtain the asked-for $\widetilde O(n + \max dj)$-time algorithm.