Approximate majorization and fair online load balancing

Abstract

This article relates the notion of fairness in online routing and load balancing to vector majorization as developed by Hardy et al. [1929]. We define α -supermajorization as an approximate form of vector majorization, and show that this definition generalizes and strengthens the prefix measure proposed by Kleinberg et al. [2001] as well as the popular notion of max-min fairness .The article revisits the problem of online load-balancing for unrelated 1-∞ machines from the viewpoint of fairness. We prove that a greedy approach is O (log n )-supermajorized by all other allocations, where n is the number of jobs. This means the greedy approach is globally O (log n )- fair . This may be contrasted with polynomial lower bounds presented by Goel et al. [2001] for fair online routing.We also define a machine-centric view of fairness using the related concept of submajorization . We prove that the greedy online algorithm is globally O (log m )- balanced , where m is the number of machines.