  We study a natural variant of scheduling that we call \emph{partial
scheduling}: In this variant an instance of a scheduling problem along with an
integer $k$ is given and one seeks an optimal schedule where not all, but only
$k$ jobs, have to be processed.
  Specifically, we aim to determine the fine-grained parameterized complexity
of partial scheduling problems parameterized by $k$ for all variants of
scheduling problems that minimize the makespan and involve unit/arbitrary
processing times, identical/unrelated parallel machines, release/due dates, and
precedence constraints. That is, we investigate whether algorithms with
runtimes of the type $f(k)n^{\mathcal{O}(1)}$ or $n^{\mathcal{O}(f(k))}$ exist
for a function $f$ that is as small as possible.
  Our contribution is two-fold: First, we categorize each variant to be either
in $\mathsf{P}$, $\mathsf{NP}$-complete and fixed-parameter tractable by $k$,
or $\mathsf{W}[1]$-hard parameterized by $k$. Second, for many interesting
cases we further investigate the run time on a finer scale and obtain run times
that are (almost) optimal assuming the Exponential Time Hypothesis. As one of
our main technical contributions, we give an $\mathcal{O}(8^kk(|V|+|E|))$ time
algorithm to solve instances of partial scheduling problems minimizing the
makespan with unit length jobs, precedence constraints and release dates, where
$G=(V,E)$ is the graph with precedence constraints.
