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.

我们考虑一个调度问题，其中有m个相同的机器并行和最小制造时间目标。最长处理时间优先 (LPT) 规则是该问题的著名近似算法。虽然理论上已经确定了它的最坏情况近似比，但众所周知，LPT 的最坏情况近似比在处理时间越短的情况下越小。我们假设每个作业的处理时间不超过给定整数k的最佳制造时间的1/ k倍。我们根据参数k和m推导出 LPT 算法的最坏情况近似比. 为此，我们将原始问题的整个实例集划分为由不同的参数k和m值定义的类。在这些类中的每一个上，我们根据参数k和m推导出最坏情况性能比的精确上限。我们还表明，存在我们的最坏情况近似比优于先前界限的实例类别。我们的界限可以在 LPT 的性能分析方面补充先前的研究。