Malleable scheduling beyond identical machines

IF 1.4 4区 工程技术 Q4 ENGINEERING, MANUFACTURING Journal of Scheduling Pub Date : 2022-05-11 DOI:10.1007/s10951-022-00733-x
Dimitris Fotakis, Jannik Matuschke, Orestis Papadigenopoulos
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引用次数: 8

Abstract

In malleable job scheduling, jobs can be executed simultaneously on multiple machines with the processing time depending on the number of allocated machines. In this setting, jobs are required to be executed non-preemptively and in unison, in the sense that they occupy, during their execution, the same time interval over all the machines of the allocated set. In this work, we study generalizations of malleable job scheduling inspired by standard scheduling on unrelated machines. Specifically, we introduce a general model of malleable job scheduling, where each machine has a (possibly different) speed for each job, and the processing time of a job j on a set of allocated machines S depends on the total speed of S with respect to j. For machines with unrelated speeds, we show that the optimal makespan cannot be approximated within a factor less than \(\frac{e}{e-1}\), unless \(P = NP\). On the positive side, we present polynomial-time algorithms with approximation ratios \(\frac{2e}{e-1}\) for machines with unrelated speeds, 3 for machines with uniform speeds, and 7/3 for restricted assignments on identical machines. Our algorithms are based on deterministic LP rounding. They result in sparse schedules, in the sense that each machine shares at most one job with other machines. We also prove lower bounds on the integrality gap of \(1+\varphi \) for unrelated speeds (\(\varphi \) is the golden ratio) and 2 for uniform speeds and restricted assignments. To indicate the generality of our approach, we show that it also yields constant factor approximation algorithms for a variant where we determine the effective speed of a set of allocated machines based on the \(L_p\) norm of their speeds.

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超越相同机器的可延展性调度
在可塑作业调度中,作业可以在多台机器上同时执行,处理时间取决于分配的机器数量。在这种设置中,作业必须以非抢占式和一致的方式执行,也就是说,它们在执行期间在分配集的所有机器上占用相同的时间间隔。在本工作中,我们研究了受不相关机器上的标准调度启发的可塑作业调度的推广。具体地说,我们引入了一个可塑作业调度的通用模型,其中每台机器对每个作业都有一个(可能不同的)速度,并且一组分配的机器S上的作业j的处理时间取决于S相对于j的总速度。对于具有不相关速度的机器,我们表明最优完工时间不能在小于\(\frac{e}{e-1}\)的因子内近似,除非\(P = NP\)。从积极的方面来看,我们提出了多项式时间算法,对于速度不相关的机器,近似比为\(\frac{2e}{e-1}\),对于速度均匀的机器,近似比为3,对于相同机器上的限制分配,近似比为7/3。我们的算法基于确定性LP舍入。它们导致稀疏调度,也就是说每台机器最多与其他机器共享一个作业。我们还证明了不相关速度(\(\varphi \)为黄金分割率)的积分差\(1+\varphi \)的下界和均匀速度和受限分配的积分差2的下界。为了表明我们的方法的通用性,我们展示了它还产生了一个变量的常数因子近似算法,其中我们根据一组分配的机器的速度的\(L_p\)范数来确定它们的有效速度。
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来源期刊
Journal of Scheduling
Journal of Scheduling 工程技术-工程:制造
CiteScore
3.80
自引率
10.00%
发文量
49
审稿时长
6-12 weeks
期刊介绍: The Journal of Scheduling provides a recognized global forum for the publication of all forms of scheduling research. First published in June 1998, Journal of Scheduling covers advances in scheduling research, such as the latest techniques, applications, theoretical issues and novel approaches to problems. The journal is of direct relevance to the areas of Computer Science, Discrete Mathematics, Operational Research, Engineering, Management, Artificial Intelligence, Construction, Distribution, Manufacturing, Transport, Aerospace and Retail and Service Industries. These disciplines face complex scheduling needs and all stand to gain from advances in scheduling technology and understanding.
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