Efficient approximation schemes for scheduling on a stochastic number of machines

Leah Epstein, Asaf Levin
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引用次数: 0

Abstract

We study three two-stage optimization problems with a similar structure and different objectives. In the first stage of each problem, the goal is to assign input jobs of positive sizes to unsplittable bags. After this assignment is decided, the realization of the number of identical machines that will be available is revealed. Then, in the second stage, the bags are assigned to machines. The probability vector of the number of machines in the second stage is known to the algorithm as part of the input before making the decisions of the first stage. Thus, the vector of machine completion times is a random variable. The goal of the first problem is to minimize the expected value of the makespan of the second stage schedule, while the goal of the second problem is to maximize the expected value of the minimum completion time of the machines in the second stage solution. The goal of the third problem is to minimize the \ell_p norm for a fixed p>1, where the norm is applied on machines' completion times vectors. Each one of the first two problems admits a PTAS as Buchem et al. showed recently. Here we significantly improve all their results by designing an EPTAS for each one of these problems. We also design an EPTAS for \ell_p norm minimization for any p>1.
随机机器数量调度的高效近似方案
我们研究了三个结构相似、目标不同的两阶段优化问题。在每个问题的第一阶段,目标都是将正大小的输入任务分配给不可拆分的包。分配决定后,将显示可提供的相同机器数量。然后,在第二阶段,将工作包分配到机器上。第二阶段中机器数量的概率向量是算法在做出第一阶段决策前已知的输入。因此,机器完成时间的向量是一个随机变量。第一个问题的目标是最小化第二阶段计划的时间跨度的期望值,而第二个问题的目标是最大化第二阶段解决方案中机器最小完成时间的期望值。第三个问题的目标是在固定的 p>1 条件下,最大限度地减小 \ell_p 准则,其中该准则应用于机器的完成时间向量。正如 Buchem 等人最近证明的那样,前两个问题中的每一个问题都有一个PTAS。在这里,我们通过为每个问题设计一个 EPTAS,大大改进了他们的所有结果。我们还为任意 p>1 的 ell_p 准则最小化问题设计了一个 EPTAS。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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