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Abstract

We consider the problem of scheduling $n$ independent jobs on $m$ unrelated parallel machines. Each job has to be processed by exactly one machine, processing job $j$ on machine $i$ requires $p_{ij}$ time units, and the objective is to minimize the makespan, i.e. the maximum job completion time. We focus on the case when $m$ is fixed and develop a fully polynomial approximation scheme whose running time depends only linearly on $n$. In the second half of the paper we extend this result to a variant of the problem, where processing job $j$ on machine $i$ also incurs a cost of $c_{ij}$, and thus there are two optimization criteria: makespan and cost. We show that for any fixed $m$, there is a fully polynomial approximation scheme that, given values $T$ and $C$, computes for any fixed $\epsilon > 0$ a schedule in $O(n)$ time with makespan at most $(1+\epsilon)T$ and cost at most $(1 + \epsilon)C$, if there exists a schedule of makespan $T$ and cost $C$.