  Scheduling problems are fundamental in combinatorial optimization. Much work
has been done on approximation algorithms for NP-hard cases, but relatively
little is known about exact solutions when some part of the input is a fixed
parameter. In 2014, Mnich and Wiese initiated a systematic study in this
direction.
  In this paper we continue this study and show that several additional cases
of fundamental scheduling problems are fixed parameter tractable for some
natural parameters. Our main tool is n-fold integer programming, a recent
variable dimension technique which we believe to be highly relevant for the
parameterized complexity community. This paper serves to showcase and highlight
this technique.
  Specifically, we show the following four scheduling problems to be
fixed-parameter tractable, where p max is the maximum processing time of a job
and w max is the maximum weight of a job:
  - Makespan minimization on uniformly related machines $(Q||C_{max} )$
parameterized by $p_{max}$,
  - Makespan minimization on unrelated machines $(R||C_{max} )$ parameterized
by $p_{max}$ and the number of kinds of machines,
  - Sum of weighted completion times minimization on unrelated machines $(R||
\sum w_i C_i )$ parameterized by $p_{max} + w_{max}$ and the number of kinds of
machines,
  - The same problem, $(R|| \sum w_i C_i),$ parameterized by the number of
distinct job times and the number of machines.
