  Fixed-parameter tractability analysis and scheduling are two core domains of
combinatorial optimization which led to deep understanding of many important
algorithmic questions. However, even though fixed-parameter algorithms are
appealing for many reasons, no such algorithms are known for many fundamental
scheduling problems.
  In this paper we present the first fixed-parameter algorithms for classical
scheduling problems such as makespan minimization, scheduling with
job-dependent cost functions-one important example being weighted flow time-and
scheduling with rejection. To this end, we identify crucial parameters that
determine the problems' complexity. In particular, we manage to cope with the
problem complexity stemming from numeric input values, such as job processing
times, which is usually a core bottleneck in the design of fixed-parameter
algorithms. We complement our algorithms with W[1]-hardness results showing
that for smaller sets of parameters the respective problems do not allow
FPT-algorithms. In particular, our positive and negative results for scheduling
with rejection explore a research direction proposed by D\'aniel Marx.
  We hope that our contribution yields a new and fresh perspective on
scheduling and fixed-parameter algorithms and will lead to further fruitful
interdisciplinary research connecting these two areas.
