Solving Optimization Problems with Diseconomies of Scale via Decoupling

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 ||Σ_i=1ⁿX_i||_q ≤ Cq ||Σ_i=1ⁿ Y_i||_q, where Xi are independent nonnegative random variables, Yi 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. However, the optimal constant Cq was not known. We show that the optimal constant is C_q = A_q^1/q.