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# Scheduling Parallel Jobs to Minimize Makespan

## Johannes, Berit

## Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin

We consider the NP-hard problem of scheduling parallel jobs with release dates on identical parallel machines to minimize the makespan. A parallel job requires simultaneously a pre-specified, job-dependent number of machines when being processed. Our main result is the following. The makespan of a (non-preemptive) schedule constructed by any listscheduling algorithm is within a factor of 2 of the optimal preemptive makespan. This gives the best known approximation algorithms for both the preemptive and the non-preemptive variant of the problem, improving upon previously known performance guarantees of 3. We also show that no listscheduling algorithm can achieve a better performance guarantee than 2 for the non-preemptive problem, no matter which priority list is chosen. Since listscheduling also works in the online setting in which jobs arrive over time and the length of a job becomes only known when it completes, the main result yields a deterministic online algorithm with competitive ratio 2 as well. In addition, we consider a different online model in which jobs arrive one by one and need to be scheduled before the next job becomes known. In this context, no listscheduling algorithm has a constant competitive ratio. We present the ﬁrst online algorithm for scheduling parallel jobs with a constant competitive ratio. We also prove a new information-theoretic lower bound of 2:25 for the competitive ratio of any deterministic online algorithm for this model.