Tight approximation bounds for the LPT rule applied to identical parallel machines with small jobs

Myungho Lee, Kangbok Lee () and Michael Pinedo
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Myungho Lee: Postech
Kangbok Lee: Postech
Michael Pinedo: New York University

Journal of Scheduling, 2022, vol. 25, issue 6, No 7, 740 pages

Abstract: 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.

Keywords: Identical parallel machine scheduling; Makespan minimization; LPT rule; Approximation algorithms; Processing time restriction (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10951-022-00742-w

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