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