Approximate Majorization And Fair Online Load Balancing

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 alpha-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-infinity 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.