### 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 (equation Presented) , where X_{i} are independent nonnegative random variables, Y_{i} are possibly dependent nonnegative random variables, and each Y_{i} has the same distribution as X_{i}. The inequality was proved by de la Peña in 1990. De la Peña, Ibragimov, and Sharakhmetov showed that C_{q} ≤ 2 for q ∈ (1, 2) and C_{q} ≤ A^{1} _{q} ^{/q} for q ≥ 2. We show that the optimal constant is C_{q} = A^{1} _{q} ^{/q} for any q ≥ 1. We then prove a more general inequality: For every convex function φ, (equation Presented) , and, for every concave function ψ, (equation Presented) , where P is a Poisson random variable with parameter 1 independent of the random variables Y_{i}

Original language | English (US) |
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Article number | 42 |

Journal | Journal of the ACM |

Volume | 65 |

Issue number | 6 |

DOIs | |

State | Published - Nov 2018 |

### Keywords

- Approximation algorithms
- Energy efficient routing
- Energy efficient scheduling
- Optimization

### ASJC Scopus subject areas

- Software
- Control and Systems Engineering
- Information Systems
- Hardware and Architecture
- Artificial Intelligence

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## Cite this

*Journal of the ACM*,

*65*(6), [42]. https://doi.org/10.1145/3266140