Minimizing the Weighted Number of Tardy Jobs is W[1]-hard
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910286475165696 |
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| author | Heeger, Klaus Hermelin, Danny |
| author_facet | Heeger, Klaus Hermelin, Danny |
| contents | We consider the $1||\sum w_J U_j$ problem, the problem of minimizing the weighted number of tardy jobs on a single machine. This problem is one of the most basic and fundamental problems in scheduling theory, with several different applications both in theory and practice. We prove that $1||\sum w_J U_j$ is W[1]-hard with respect to the number $p_{\#}$ of different processing times in the input, as well as with respect to the number $w_{\#}$ of different weights in the input. This, along with previous work, provides a complete picture for $1||\sum w_J U_j$ from the perspective of parameterized complexity, as well as almost tight complexity bounds for the problem under the Exponential Time Hypothesis (ETH). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_01740 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Minimizing the Weighted Number of Tardy Jobs is W[1]-hard Heeger, Klaus Hermelin, Danny Data Structures and Algorithms Computational Complexity We consider the $1||\sum w_J U_j$ problem, the problem of minimizing the weighted number of tardy jobs on a single machine. This problem is one of the most basic and fundamental problems in scheduling theory, with several different applications both in theory and practice. We prove that $1||\sum w_J U_j$ is W[1]-hard with respect to the number $p_{\#}$ of different processing times in the input, as well as with respect to the number $w_{\#}$ of different weights in the input. This, along with previous work, provides a complete picture for $1||\sum w_J U_j$ from the perspective of parameterized complexity, as well as almost tight complexity bounds for the problem under the Exponential Time Hypothesis (ETH). |
| title | Minimizing the Weighted Number of Tardy Jobs is W[1]-hard |
| topic | Data Structures and Algorithms Computational Complexity |
| url | https://arxiv.org/abs/2401.01740 |