The Art of Staying Ahead of Deadlines: Improved Algorithms for the Minimum Tardy Processing Time
<|reference_start|>The Art of Staying Ahead of Deadlines: Improved Algorithms for the Minimum Tardy Processing Time: 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.<|reference_end|>
@article{stoian2024the,
    title={The Art of Staying Ahead of Deadlines: Improved Algorithms for the
  Minimum Tardy Processing Time},
    author={Mihail Stoian},
    journal={arXiv preprint arXiv:2402.16847},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.16847},
    primaryClass={cs.DS cs.CC}
}