# Tight Approximation Algorithms for Scheduling with Fixed Jobs and Non-Availability

We study two closely related problems in non-preemptive scheduling of sequential jobs on identical parallel machines. In these two settings there are either fixed jobs or non-availability intervals during which the machines are not available; in both cases, the objective is to minimize the makespan. Both formulations have different applications, e.g. in turnaround scheduling or overlay computing. For both problems we contribute approximation algorithms with an improved ratio of $3/2+\epsilon$, respectively, which we refine to approximation algorithms with ratio $3/2$. For scheduling with fixed jobs, a lower bound of $3/2$ on the approximation ratio has been obtained by Scharbrodt, Steger & Weisser: for scheduling with non-availability we provide the same lower bound. In total, our approximation ratio for both problems is tight via suitable inapproximability results. We use dual approximation, creation of a gap structure and job configurations, and a PTAS for the multiple subset sum problem. However, the main feature of our algorithms is a new technique for the assignment of large jobs via flexible rounding. Our new technique is based on an interesting cyclic shifting argument in combination with a network flow model for the assignment of jobs to large gaps.

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