Scheduling on (Un-)Related Machines with Setup Times

TL;DR

This paper studies scheduling jobs on machines with setup times when switching between job classes, providing approximation algorithms for related and unrelated machines, including a polynomial-time scheme for related machines.

Contribution

It introduces approximation algorithms for scheduling with setup times on related and unrelated machines, including a PTAS for related machines and an optimal approximation for unrelated machines.

Findings

Polynomial-time approximation scheme for uniformly related machines.

O(log n + log m)-approximation for unrelated machines, proven optimal.

Two special cases with constant factor approximations.

Abstract

We consider a natural generalization of scheduling $n$ jobs on $m$ parallel machines so as to minimize the makespan. In our extension the set of jobs is partitioned into several classes and a machine requires a setup whenever it switches from processing jobs of one class to jobs of a different class. During such a setup, a machine cannot process jobs and the duration of a setup may depend on the machine as well as the class of the job to be processed next. For this problem, we study approximation algorithms for non-identical machines. We develop a polynomial-time approximation scheme for uniformly related machines. For unrelated machines we obtain an $O(\log n + \log m)$-approximation, which we show to be optimal (up to constant factors) unless $NP \subset RP$. We also identify two special cases that admit constant factor approximations.