A Fully Polynomial-Time Approximation Scheme for Speed Scaling with a Sleep State

We study classical deadline-based preemptive scheduling of jobs in a computing environment equipped with both dynamic speed scaling and sleep state capabilities: Each job is specified by a release time, a deadline and a processing volume, and has to be scheduled on a single, speed-scalable processor...

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Published in:	Algorithmica Vol. 81; no. 9; pp. 3725 - 3745
Main Authors:	Antoniadis, Antonios, Huang, Chien-Chung, Ott, Sebastian
Format:	Journal Article
Language:	English
Published:	New York Springer US 16.09.2019 Springer Nature B.V Springer Verlag
Subjects:	Algorithm Analysis and Problem Complexity Algorithms Approximation Complexity Computational Geometry Computer Science Computer Systems Organization and Communication Networks Data Structures and Algorithms Data Structures and Information Theory Energy consumption Lower bounds Mathematical analysis Mathematics of Computing Microprocessors Polynomials Preempting Production scheduling Scaling Schedules Sleep Theory of Computation Approximation algorithms Polynomial-time approximation scheme Energy efficiency
ISSN:	0178-4617, 1432-0541
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Abstract	We study classical deadline-based preemptive scheduling of jobs in a computing environment equipped with both dynamic speed scaling and sleep state capabilities: Each job is specified by a release time, a deadline and a processing volume, and has to be scheduled on a single, speed-scalable processor that is supplied with a sleep state. In the sleep state, the processor consumes no energy, but a constant wake-up cost is required to transition back to the active state. In contrast to speed scaling alone, the addition of a sleep state makes it sometimes beneficial to accelerate the processing of jobs in order to transition the processor to the sleep state for longer amounts of time and incur further energy savings. The goal is to output a feasible schedule that minimizes the energy consumption. Since the introduction of the problem by Irani et al. (ACM Trans Algorithms 3(4), 2007 ), its exact computational complexity has been repeatedly posed as an open question (see e.g. Albers and Antoniadis in ACM Trans Algorithms 10(2):9, 2014 ; Baptiste et al. in ACM Trans Algorithms 8(3):26, 2012 ; Irani and Pruhs in SIGACT News 36(2):63–76, 2005 ). The currently best known upper and lower bounds are a 4 / 3-approximation algorithm and NP-hardness due to Albers and Antoniadis ( 2014 ) and Kumar and Shannigrahi (CoRR, 2013 . arXiv:1304.7373 ), respectively. We close the aforementioned gap between the upper and lower bound on the computational complexity of speed scaling with sleep state by presenting a fully polynomial-time approximation scheme for the problem. The scheme is based on a transformation to a non-preemptive variant of the problem, and a discretization that exploits a carefully defined lexicographical ordering among schedules.
AbstractList	We study classical deadline-based preemptive scheduling of jobs in a computing environment equipped with both dynamic speed scaling and sleep state capabilities: Each job is specified by a release time, a deadline and a processing volume, and has to be scheduled on a single, speed-scalable processor that is supplied with a sleep state. In the sleep state, the processor consumes no energy, but a constant wake-up cost is required to transition back to the active state. In contrast to speed scaling alone, the addition of a sleep state makes it sometimes beneficial to accelerate the processing of jobs in order to transition the processor to the sleep state for longer amounts of time and incur further energy savings. The goal is to output a feasible schedule that minimizes the energy consumption. Since the introduction of the problem by Irani et al. (ACM Trans Algorithms 3(4), 2007), its exact computational complexity has been repeatedly posed as an open question (see e.g. Albers and Antoniadis in ACM Trans Algorithms 10(2):9, 2014; Baptiste et al. in ACM Trans Algorithms 8(3):26, 2012; Irani and Pruhs in SIGACT News 36(2):63–76, 2005). The currently best known upper and lower bounds are a 4 / 3-approximation algorithm and NP-hardness due to Albers and Antoniadis (2014) and Kumar and Shannigrahi (CoRR, 2013. arXiv:1304.7373), respectively. We close the aforementioned gap between the upper and lower bound on the computational complexity of speed scaling with sleep state by presenting a fully polynomial-time approximation scheme for the problem. The scheme is based on a transformation to a non-preemptive variant of the problem, and a discretization that exploits a carefully defined lexicographical ordering among schedules. We study classical deadline-based preemptive scheduling of jobs in a computing environment equipped with both dynamic speed scaling and sleep state capabilities: Each job is specified by a release time, a deadline and a processing volume, and has to be scheduled on a single, speed-scalable processor that is supplied with a sleep state. In the sleep state, the processor consumes no energy, but a constant wake-up cost is required to transition back to the active state. In contrast to speed scaling alone, the addition of a sleep state makes it sometimes beneficial to accelerate the processing of jobs in order to transition the processor to the sleep state for longer amounts of time and incur further energy savings. The goal is to output a feasible schedule that minimizes the energy consumption. Since the introduction of the problem by Irani et al. (ACM Trans Algorithms 3(4), 2007 ), its exact computational complexity has been repeatedly posed as an open question (see e.g. Albers and Antoniadis in ACM Trans Algorithms 10(2):9, 2014 ; Baptiste et al. in ACM Trans Algorithms 8(3):26, 2012 ; Irani and Pruhs in SIGACT News 36(2):63–76, 2005 ). The currently best known upper and lower bounds are a 4 / 3-approximation algorithm and NP-hardness due to Albers and Antoniadis ( 2014 ) and Kumar and Shannigrahi (CoRR, 2013 . arXiv:1304.7373 ), respectively. We close the aforementioned gap between the upper and lower bound on the computational complexity of speed scaling with sleep state by presenting a fully polynomial-time approximation scheme for the problem. The scheme is based on a transformation to a non-preemptive variant of the problem, and a discretization that exploits a carefully defined lexicographical ordering among schedules.
Author	Huang, Chien-Chung Antoniadis, Antonios Ott, Sebastian
Author_xml	– sequence: 1 givenname: Antonios orcidid: 0000-0003-2152-7883 surname: Antoniadis fullname: Antoniadis, Antonios email: aantonia@mpi-inf.mpg.de organization: Saarland University and Max-Planck-Institut für Informatik – sequence: 2 givenname: Chien-Chung surname: Huang fullname: Huang, Chien-Chung organization: École Normale Supérieure – sequence: 3 givenname: Sebastian surname: Ott fullname: Ott, Sebastian organization: Max-Planck-Institut für Informatik
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Approximation Complexity Computational Geometry Computer Science Computer Systems Organization and Communication Networks Data Structures and Algorithms Data Structures and Information Theory Energy consumption Lower bounds Mathematical analysis Mathematics of Computing Microprocessors Polynomials Preempting Production scheduling Scaling Schedules Sleep Theory of Computation
Title	A Fully Polynomial-Time Approximation Scheme for Speed Scaling with a Sleep State
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