Approximation Algorithms in Batch Processing

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We study the scheduling of a set of jobs, each characterised by a release (arrival) time and a processing time, for a batch processing machine capable of running (at most) a fixed number of jobs at a time. When the job release times and processing times are known a-priori and the inputs are integers, we obtained an algorithm for finding a schedule with the minimum makespan. The running time is pseudo-polynomial when the number of distinct job release times is constant. We also ob- tained a fully polynomial time approximation scheme when the number of distinct job release times is constant, and a polynomial time approxi- mation scheme when that number is arbitrary. When nothing is known about a job until it arrives, i.e., the on-line setting, we proved a lower bound of \( (\sqrt 5 + 1)/2 \) on the competitive ratio of any approximation al- gorithm. This bound is tight when the machine capacity is unbounded.