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

• Published in: Journal of scheduling Vol. 25; no. 6; pp. 721 - 740
• Main Authors: Lee, Myungho, Lee, Kangbok, Pinedo, Michael
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2022; Springer Nature B.V
• Subjects: Algorithms; Approximation; Artificial Intelligence; Business and Management; Calculus of Variations and Optimal Control; Optimization; Mathematical analysis; Operations Research/Decision Theory; Optimization; Parameters; Supply Chain Management; Upper bounds; Identical parallel machine scheduling; Processing time restriction; Approximation algorithms; Makespan minimization; LPT rule
• ISSN: 1094-6136; 1099-1425
• DOI: 10.1007/s10951-022-00742-w

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.