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=1 n X i || q ≤ Cq ||Σ i=1 n 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 .