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A Fast Distributed Algorithm for αα-Fair Packing Problems

Abstract

Over the past two decades, fair resource allocation problems have received considerable attention in a variety of application areas. While polynomial time distributed algorithms have been designed for max-min fair resource allocation, the design of distributed algorithms with convergence guarantees for the more general α\alpha-fair allocations received little attention. In this paper, we study weighted α\alpha-fair packing problems, that is, the problems of maximizing the objective functions jwjxj1α/(1α)\sum_j w_j x_j^{1-\alpha}/(1-\alpha) when α>0\alpha > 0, α1\alpha \neq 1 and jwjlnxj\sum_j w_j \ln x_j when α=1\alpha = 1 over linear constraints AxbAx \leq b, x0x\geq 0, where wjw_j are positive weights and AA and bb are non-negative. We consider the distributed computation model that was used for packing linear programs and network utility maximization problems. Under this model, we provide a distributed algorithm for general α\alpha. The algorithm uses simple local update rules and is stateless (namely, it allows asynchronous updates, is self-stabilizing, and allows incremental and local adjustments). The algorithm converges to approximate solutions in running times that have an inverse polynomial dependence on the approximation parameter ε\varepsilon and poly-logarithmic dependence on the problem size. These are the best convergence times known for the α\alpha-fair packing problems.

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