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A Fast Distributed Algorithm for αα-Fair Packing Problems

Abstract

Over the past two decades, fair resource allocation problems 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 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). It converges to approximate solutions in running times that have an inverse polynomial dependence on the approximation parameter ε\varepsilon. The convergence time has polylogarithmic dependence on the problem size for α1\alpha \neq 1, and a nearly-linear dependence on the number of variables for α=1\alpha = 1. These are the best convergence times known for these problems.

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