The Art of Staying Ahead of Deadlines: Improved Algorithms for the Minimum Tardy Processing Time
| Autor: | Stoian, Mihail |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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. Comment: The runtime analysis is incorrect: the contribution of the processing times in [d_j - p_j, d_{j - 1}] is not taken into account. We thank N. Fischer for pointing this out |
| Databáze: | arXiv |
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