## Abstract

We consider a natural online optimization problem set on the real line. The state of the online algorithm at each integer time t is a location x_{t} on the real line. At each integer time t, a convex function ft(x) arrives online. In response, the online algorithm picks a new location x_{t}. The cost paid by the online algorithm for this response is the distance moved, namely |x_{t} - x_{t-1}|, plus the value of the function at the final destination, namely Σ_{t}(x_{t}). The objective is then to minimize the aggregate cost over all time, namely Σ_{t} (|x_{t} - x_{t-1}| + f_{t}(x_{t})). The motivating application is rightsizing power-proportional data centers. We give a 2-competitive algorithm for this problem. We also give a 3-competitive memoryless algorithm, and show that this is the best competitive ratio achievable by a deterministic memoryless algorithm. Finally we show that this online problem is strictly harder than the standard ski rental problem.

Original language | English |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 18th International Workshop, APPROX 2015, and 19th International Workshop, RANDOM 2015 |

Editors | Naveen Garg, Klaus Jansen, Anup Rao, Jose D. P. Rolim |

Publisher | Schloss Dagstuhl - Leibniz-Zentrum für Informatik |

Pages | 96-109 |

Number of pages | 14 |

ISBN (Electronic) | 9783939897897 |

DOIs | |

Publication status | Published - 1 Aug 2015 |

Event | 18th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2015, and 19th International Workshop on Randomization and Computation, RANDOM 2015 - Princeton, United States Duration: 24 Aug 2015 → 26 Aug 2015 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 40 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 18th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2015, and 19th International Workshop on Randomization and Computation, RANDOM 2015 |
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Country/Territory | United States |

City | Princeton |

Period | 24/08/15 → 26/08/15 |

### Bibliographical note

Publisher Copyright:© Nikhil Bansal, Anupam Gupta, Ravishankar Krishnaswamy, Kirk Pruhs, Kevin Schewior, and Cliff Stein.

## Keywords

- Scheduling
- Stochastic