Near-Linear Approximation Algorithms for Scheduling Problems with Batch Setup Times

Abstract

We investigate the scheduling of $n$ jobs divided into $c$ classes on $m$ identical parallel machines. For every class there is a setup time which is required whenever a machine switches from the processing of one class to another class. The objective is to find a schedule that minimizes the makespan. We give near-linear approximation algorithms for the following problem variants: the non-preemptive context where jobs may not be preempted, the preemptive context where jobs may be preempted but not parallelized, as well as the splittable context where jobs may be preempted and parallelized. We present the first algorithm improving the previously best approximation ratio of $2$ to a better ratio of $3/2$ in the preemptive case. In more detail, for all three flavors we present an approximation ratio $2$ with running time $\mathcal{O}(n)$, ratio $3/2+\varepsilon$ in time $\mathcal{O}(n\log 1/\varepsilon)$ as well as a ratio of $3/2$. The $(3/2)$-approximate algorithms have different running times. In the non-preemptive case we get time $\mathcal{O}(n\log (n+\Delta))$ where $\Delta$ is the largest value of the input. The splittable approximation runs in time $\mathcal{O}(n+c\log(c+m))$ whereas the preemptive algorithm has a running time $\mathcal{O}(n \log (c+m)) \leq \mathcal{O}(n \log n)$. So far, no PTAS is known for the preemptive problem without restrictions, so we make progress towards that question. Recently Jansen et al. found an EPTAS for the splittable and non-preemptive case but with impractical running times exponential in $1/\varepsilon$.