### Abstract

We consider the online machine minimization problem in which jobs with hard deadlines arrive online over time at their release dates. The task is to determine a feasible preemptive schedule on a minimum number of machines. Our main result is a general O(log m)-competitive algorithm for the online problem, where m is the optimal number of machines used in an offline solution. This is the first improvement to an intriguing problem in nearly two decades. To date, the best known result is a O(log(pmax/p _{min} ))-competitive algorithm by Phillips et al. [Optimal time-critical scheduling via resource augmentation, STOC, 1997] that depends on the ratio of maximum and minimum job sizes, pmax and pmin. Even for m = 2 no better algorithm was known. Our algorithm is in this case constant-competitive. When applied to laminar or agreeable instances, our algorithm achieves a competitive ratio of O(1) even independently of m. The following two key components lead to our new result. First, we derive a new lower bound on the optimum value that relates the laxity and the number of jobs with intersecting time windows. Then, we design a new algorithm that is tailored to this lower bound and balances the delay of jobs by taking the number of currently running jobs into account.

Original language | English |
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Pages (from-to) | 2057-2077 |

Number of pages | 21 |

Journal | SIAM Journal on Computing |

Volume | 47 |

Issue number | 6 |

DOIs | |

State | Published - Jan 1 2018 |

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### Keywords

- Competitive analysis
- Multiprocessor scheduling
- Online algorithms

### Cite this

*SIAM Journal on Computing*,

*47*(6), 2057-2077. https://doi.org/10.1137/17M116032X