## 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. [17], its exact computational complexity has been repeatedly posed as an open question (see e.g. [2,9,16]). The currently best known upper and lower bounds are a 4/3-approximation algorithm and NP-hardness due to [2] and [2,18], 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 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.

Original language | English |
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Title of host publication | Proceedings of the Twenty-Sixth Annual Symposium on Discrete Algorithms 2015 |

Editors | Piotr Indyk |

Publisher | SIAM |

Number of pages | 12 |

ISBN (Electronic) | 978-1-61197-373-0 |

ISBN (Print) | 978-1-61197-374-7 |

DOIs | |

Publication status | Published - 2015 |

Externally published | Yes |

Event | 26th Annual ACM-SIAM Symposium on Discrete Algorithms 2015 - San Diego, United States Duration: 4 Jan 2015 → 6 Jan 2015 Conference number: 26 |

### Conference

Conference | 26th Annual ACM-SIAM Symposium on Discrete Algorithms 2015 |
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Country/Territory | United States |

City | San Diego |

Period | 4/01/15 → 6/01/15 |