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arXiv · 2401.01740

Minimizing the Weighted Number of Tardy Jobs is W[1]-hard

Klaus Heeger·Danny Hermelin
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Abstract

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).

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Klaus Heeger, Danny Hermelin. 2024-01-03. Minimizing the Weighted Number of Tardy Jobs is W[1]-hard. https://arxiv.org/abs/2401.01740

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