This paper studies Makespan Minimization in the secretary model. Formally, jobs, specified by their processing times, are presented in a uniformly random order. An online algorithm has to assign each job permanently and irrevocably to one of m parallel and identical machines such that the expected time it takes to process them all, the makespan, is minimized. We give two deterministic algorithms. First, a straightforward adaptation of the semi-online strategy LightLoad provides a very simple algorithm retaining its competitive ratio of 1.75. A new and sophisticated algorithm is 1.535-competitive. These competitive ratios are not only obtained in expectation but, in fact, for all but a very tiny fraction of job orders. Classically, online makespan minimization only considers the worst-case order. Here, no competitive ratio below 1.885 for deterministic algorithms and 1.581 using randomization is possible. The best randomized algorithm so far is 1.916-competitive. Our results show that classical worst-case orders are quite rare and pessimistic for many applications. They also demonstrate the power of randomization when compared to much stronger deterministic reordering models. We complement our results by providing first lower bounds. A competitive ratio obtained on nearly all possible job orders must be at least 1.257. This implies a lower bound of 1.043 for both deterministic and randomized algorithms in the general model.
翻译:纸质研究在秘书模型中实现了最小化。 形式上, 由处理时间指定的工作, 以统一的随机顺序显示。 在线算法必须永久和不可撤销地将每份工作指派给一个平行和完全相同的机器, 以便尽可能缩短处理它们所需的时间。 我们给出了两种确定式算法。 首先, 直接调整半在线战略 LightLoad 提供了一种非常简单的算法, 保留其1. 75 的竞争性比率。 一个新的和尖端的算法具有1.535的竞争力。 这些竞争性比率不仅在预期中获得, 事实上, 对所有工作订单来说, 这些竞争性比率非常小。 典型的在线最小化只考虑最坏的顺序。 在这里, 确定性算法没有低于1.885的竞争性比率, 使用随机化的1.581 。 目前, 最佳随机化的算法是1. 916 具有竞争力。 我们的结果表明, 典型的最坏的算法非常罕见和悲观的算法对于许多应用来说都是非常罕见的。 它们也显示了随机化的力量, 与较强的确定性重的重新排序模型相比, 事实上, 也是非常小的一部分。 典型的最小的最小的最小的最小的最小的最小的排序。 我们的排序必须以最低的排序法 。