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Abstract:
We consider the problem of preemptively scheduling a set of $n$ jobs
on $m$ (identical, uniformly related, or unrelated) parallel
machines. The scheduler may reject a subset of the jobs and thereby
incur job-dependent penalties for each rejected job, and he must
construct a schedule for the remaining jobs so as to optimize the
preemptive makespan on the $m$ machines plus the sum of the
penalties of the jobs rejected.
We provide a complete classification of these scheduling problems
with respect to complexity and approximability. Our main results
are on the variant with an arbitrary number of unrelated machines.
This variant is \apx-hard, and we design a $1.58$-approximation
algorithm for it. All other considered variants are weakly
\np-hard, and we provide fully polynomial time approximation schemes
for them. Finally, we argue that our results for unrelated machines
can be carried over to the corresponding preemptive open shop
scheduling problem with rejection.
