Approximate majorization and fair online load balancing

Authors:
Ashish Goel;Adam Meyerson;Serge Plotkin
Affiliations:
Stanford University, Stanford, CA;University of California, Los Angeles, Los Angeles, CA;Stanford University, Stanford, CA
Venue:
ACM Transactions on Algorithms (TALG)
Year:
2005

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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.