Konstantin Makarychev (Northwestern University)
Monday, March 13, 2017, 2:00pm
Title: Solving optimization problems with a diseconomy of scale
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, 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 several optimization problems with a diseconomy of scale such as 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. In the Minimum Energy Efficient Routing problem, the goal is to route an unsplittable multicommodity flow in a network/graph between terminals so as to minimize the energy consumption on the links/edges.
Our analysis relies on a decoupling inequality for nonnegative random variables. Consider arbitrary nonnegative jointly distributed random variables Y1,…,Yn. Let X1,…,Xn be independent copies of Y1,…,Yn such that all Xi are independent and each Xi has the same distribution as Yi. Then, E(X1+…+Xn)q < Cq E(Y1+…+Yn)q. The inequality was proved by de la Pena in 1990. However, the optimal constant Cq was not known. We show that the optimal constant is Cq=Aq.
This is a joint work with Maxim Sviridenko, Yahoo Labs.