In this work, we study the computational (parameterized) complexity of $P \mid r_j, p_j=p \mid \sum_j w_j U_j$. Here, we are given $m$ identical parallel machines and $n$ jobs with equal processing time, each characterized by a release date, a due date, and a weight. The task is to find a feasible schedule, that is, an assignment of the jobs to starting times on machines, such that no job starts before its release date and no machine processes several jobs at the same time, that minimizes the weighted number of tardy jobs. A job is considered tardy if it finishes after its due date. Our main contribution is showing that $P \mid r_j, p_j=p \mid \sum_j U_j$ (the unweighted version of the problem) is NP-hard and W[2]-hard when parameterized by the number of machines. The former resolves an open problem in Note 2.1.19 by Kravchenko and Werner [Journal of Scheduling, 2011] and Open Problem 2 by Sgall [ESA, 2012], and the latter resolves Open Problem 7 by Mnich and van Bevern [Computers & Operations Research, 2018]. Furthermore, our result shows that the known XP-algorithm for $P \mid r_j, p_j=p \mid \sum_j w_j U_j$ parameterized by the number of machines is optimal from a classification standpoint. On the algorithmic side, we provide alternative running time bounds for the above-mentioned known XP-algorithm. Our analysis shows that $P \mid r_j, p_j=p \mid \sum_j w_j U_j$ is contained in XP when parameterized by the processing time, and that it is contained in FPT when parameterized by the combination of the number of machines and the processing time. Finally, we give an FPT-algorithm for $P \mid r_j, p_j=p \mid \sum_j w_j U_j$ parameterized by the number of release dates or the number of due dates. With this work, we lay out the foundation for a systematic study of the parameterized complexity of $P \mid r_j, p_j=p \mid \sum_j w_j U_j$.

翻译：本文研究了计算（参数化）复杂性 $P \mid r_j, p_j=p \mid \sum_j w_j U_j$。给定 $m$ 台相同并行机和 $n$ 个具有相同加工时间的工件，每个工件具有释放时间、截止时间和权重。任务是为工件分配在机器上的开始时间，使得不存在在释放时间前开始的工件，且同一时刻每台机器至多处理一个工件，同时最小化加权误工工件数量。若工件在截止时间后完成，则视为误工。主要贡献在于：当以机器数为参数时，我们证明了 $P \mid r_j, p_j=p \mid \sum_j U_j$（该问题的未加权版本）是 NP-难且 W[2]-难的。前者解决了 Kravchenko 和 Werner [Journal of Scheduling, 2011] 的 Note 2.1.19 中的开放问题与 Sgall [ESA, 2012] 的开放问题2，后者解决了 Mnich 和 van Bevern [Computers & Operations Research, 2018] 的开放问题7。此外，该结果表明已知的以机器数为参数的 $P \mid r_j, p_j=p \mid \sum_j w_j U_j$ XP-算法在分类意义上是最优的。在算法方面，我们为上述已知的 XP-算法提供了替代运行时间界。分析表明：以加工时间为参数时，$P \mid r_j, p_j=p \mid \sum_j w_j U_j$ 属于 XP 类；而以机器数与加工时间的组合为参数时，属于 FPT 类。最后，我们给出了以释放日期数或截止日期数为参数的 $P \mid r_j, p_j=p \mid \sum_j w_j U_j$ 的 FPT-算法。本文为系统研究 $P \mid r_j, p_j=p \mid \sum_j w_j U_j$ 的参数复杂性奠定了基础。