Solving Optimization Problems with Diseconomies of Scale


December 18, 2014


Konstantin Makarychev




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. 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.


Konstantin Makarychev

Kostya Makarychev is a researcher in the Theory group at MSR. He obtained his PhD from Princeton in 2007, with advisor Moses Charikar. His research centers on approximation algorithms and combinatorial optimization, in particular, semidefinite programming and connections to functional analysis and metric geometry. For more details, see