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A Fast Distributed Stateless Algorithm for $α$-Fair Packing Problems

Over the past two decades, fair resource allocation problems have received considerable attention in a variety of application areas. However, little progress has been made in the design of distributed algorithms with convergence guarantees for general and commonly used $α$-fair allocations. In this paper, we study weighted $α$-fair packing problems, that is, the problems of maximizing the objective functions (i) $\sum_j w_j x_j^{1-α}/(1-α)$ when $α> 0$, $α\neq 1$ and (ii) $\sum_j w_j \ln x_j$ when $α= 1$, over linear constraints $Ax \leq b$, $x\geq 0$, where $w_j$ are positive weights and $A$ and $b$ 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 $α$ that converges to an $\varepsilon-$approximate solution in time (number of distributed iterations) that has an inverse polynomial dependence on the approximation parameter $\varepsilon$ and poly-logarithmic dependence on the problem size. This is the first distributed algorithm for weighted $α-$fair packing with poly-logarithmic convergence in the input size. The algorithm uses simple local update rules and is stateless (namely, it allows asynchronous updates, is self-stabilizing, and allows incremental and local adjustments). We also obtain a number of structural results that characterize $α-$fair allocations as the value of $α$ is varied. These results deepen our understanding of fairness guarantees in $α-$fair packing allocations, and also provide insight into the behavior of $α-$fair allocations in the asymptotic cases $α\rightarrow 0$, $α\rightarrow 1$, and $α\rightarrow \infty$.