Minimizing the Number of Tardy Jobs with Uniform Processing Times on Parallel Machines

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 $P \mid r_j, p_j=p \mid \sum_j w_j U_j$.