##### Width-Independence Beyond Linear Objectives: Distributed Fair Packing and Covering Algorithms
In network routing and resource allocation, $\alpha$-fair utility functions are concave objective functions used to model different notions of fairness in a single, generic framework. Different choices of the parameter $\alpha$ give rise to different notions of fairness, including max-min fairness ($\alpha = \infty$), proportional fairness ($\alpha=1$), and the unfair linear optimization ($\alpha = 0)$. In this work, we consider $\alpha$-fair resource allocation problems, defined as the maximization of $\alpha$-fair utility functions under packing constraints. We give improved distributed algorithms for constructing $\epsilon$-approximate solutions to such problems. Our algorithms are width-independent, i.e., their running time depends only poly-logarithmically on the largest entry of the constraint matrix, and closely matches the state-of-the-art guarantees for distributed algorithms for packing linear programs, i.e., for the case $\alpha = 0.$ The only previously known width-independent algorithms for $\alpha$-fair resource allocation, by Marasevic, Stein, and Zussman, obtained convergence times that exhibited much worse dependence on $\epsilon$ and $\alpha$ and relied on a less principled analysis. By contrast, our analysis leverages the Approximate Duality Gap framework of Diakonikolas and Orecchia to obtain better algorithms with a (slightly) simpler analysis. Finally, we introduce a natural counterpart of $\alpha$-fairness for minimization problems and motivate its usage in the context of fair task allocation. This generalization yields $\alpha$-fair covering problems, for which we provide the first width-independent nearly-linear-time approximate solvers by reducing their analysis to the $\alpha < 1$ case of the $\alpha$-fair packing problem.
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08/08/18 05:54PM
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