  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).
