### Abstract

We present a new framework for solving optimization problems with a diseconomy of scale. In such problems, our goal is to minimize the cost of resources used to perform a certain task. The cost of resources grows superlinearly, as x^{q}, q ≥ 1, with the amount x of resources used. We define a novel linear programming relaxation for such problems, and then show that the integrality gap of the relaxation is A_{q}, where A_{q} is the q-th moment of the Poisson random variable with parameter 1. Using our framework, we obtain approximation algorithms for the Minimum Energy Efficient Routing, Minimum Degree Balanced Spanning Tree, Load Balancing on Unrelated Parallel Machines, and Unrelated Parallel Machine Scheduling with Nonlinear Functions of Completion Times problems. Our analysis relies on the decoupling inequality for nonnegative random variables. The inequality states that ||σ^{n} _{i=1} Xi||q ≤ Cq ||σ^{n}_{i=1} Yi||q, where Xi are independent nonnegative random variables, Yi are possibly dependent nonnegative random variable, and each Yi has the same distribution as Xi. The inequality was proved by de la Pe;a in 1990. However, the optimal constant Cq was not known. We show that the optimal constant is C_{q} = A^{q1}/_{q}.

Original language | English (US) |
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Title of host publication | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |

Publisher | IEEE Computer Society |

Pages | 571-580 |

Number of pages | 10 |

ISBN (Electronic) | 9781479965175 |

DOIs | |

State | Published - Dec 7 2014 |

Event | 55th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2014 - Philadelphia, United States Duration: Oct 18 2014 → Oct 21 2014 |

### Publication series

Name | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |
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ISSN (Print) | 0272-5428 |

### Other

Other | 55th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2014 |
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Country | United States |

City | Philadelphia |

Period | 10/18/14 → 10/21/14 |

### ASJC Scopus subject areas

- Computer Science(all)

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## Cite this

*Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS*(pp. 571-580). [6979042] (Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS). IEEE Computer Society. https://doi.org/10.1109/FOCS.2014.67