Optimization Problems with Diseconomies of Scale via Decoupling

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 $x^q$, $q\ge 1$, 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 $$\big \|\sum_{i=1}^n X_i\big\|_{q} \leq C_q \,\big \|\sum_{i=1}^n Y_i\big\|_{q},$$ where $X_i$ are independent nonnegative random variables, $Y_i$ are possibly dependent nonnegative random variable, 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 (2003) showed that $C_q\leq 2$ for $q\in (1,2)$ and $C_q\leq A_q^{1/q}$ for $q\geq 2$. We show that the optimal constant is $C_q=A_q^{1/q}$ for any $q\geq 1$. We then prove a more general inequality: For every convex function $φ$, $$\mathbb{E}[φ\Big(\sum_{i=1}^n X_i\Big)]\leq \mathbb{E}[φ\Big(P\sum_{i=1}^n Y_i\Big)],$$ and, for every concave function $ψ$, $$\mathbb{E}[ψ\Big(\sum_{i=1}^n X_i\Big)] \geq \mathbb{E}[ψ\Big(P\sum_{i=1}^n Y_i\Big)],$$ where $P$ is a Poisson random variable with parameter 1 independent of the random variables $Y_i$.