Approximate majorization and fair online load balancing

This article relates the notion of fairness in online routing and load balancing to <i>vector majorization</i> as developed by Hardy et al. [1929]. We define α<i>-supermajorization</i> 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 <i>max-min fairness</i>.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 <i>O</i>(log <i>n</i>)-supermajorized by all other allocations, where <i>n</i> is the number of jobs. This means the greedy approach is globally <i>O</i>(log <i>n</i>)-<i>fair</i>. 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 <i>submajorization</i>. We prove that the greedy online algorithm is globally <i>O</i>(log <i>m</i>)-<i>balanced</i>, where <i>m</i> is the number of machines.

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