Abstract
We consider a scheduling problem with m identical machines in parallel and the minimum makespan objective. The Longest Processing Time first (LPT) rule is a well-known approximation algorithm for this problem. Although its worst-case approximation ratio has been determined theoretically, it is known that the worst-case approximation ratio of LPT can be smaller with instances of smaller processing times. We assume that each job’s processing time is not longer than 1/k times the optimal makespan for a given integer k. We derive the worst-case approximation ratio of the LPT algorithm in terms of parameters k and m. For that purpose, we divide the whole set of instances of the original problem into classes defined by different values of parameters k and m. On each of those classes, we derive an exact upper bound on the worst-case performance ratio as a function of parameters k and m. We also show that there exist classes of instances for which our worst-case approximation ratio is better than previous bounds. Our bound can complement previous research in terms of the performance analysis of LPT.
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The authors would like to thank anonymous referees who provided very constructive and detailed comments on a previous version of the manuscript.
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Lee, M., Lee, K. & Pinedo, M. Tight approximation bounds for the LPT rule applied to identical parallel machines with small jobs. J Sched 25, 721–740 (2022). https://doi.org/10.1007/s10951-022-00742-w
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DOI: https://doi.org/10.1007/s10951-022-00742-w