Solving optimization problems with diseconomies of scale via decoupling

K. Makarychev, M. Sviridenko

Research output: Contribution to journalArticle

1 Scopus citations

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 xq, 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 Aq, where Aq 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 Xi are independent nonnegative random variables, Yi are possibly dependent nonnegative random variables, and each Yi has the same distribution as Xi. The inequality was proved by de la Peña in 1990. De la Peña, Ibragimov, and Sharakhmetov showed that Cq ≤ 2 for q ∈ (1, 2) and Cq ≤ A1 q /q for q ≥ 2. We show that the optimal constant is Cq = A1 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 Yi

Original languageEnglish (US)
Article number42
JournalJournal of the ACM
Volume65
Issue number6
DOIs
StatePublished - 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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