Publication

A Unified Framework for Max-Min and Min-Max Fairness with Applications

Jean-Yves Le Boudec
2002
Report or working paper
Abstract

Max-min fairness is widely used in various areas of networking. In every case where it is used, there is a proof of existence and one or several algorithms for computing the max-min fair allocation; in most, but not all cases, they are based on the notion of bottlenecks. In spite of this wide applicability, there are still examples, arising in the context of mobile or peer-to-peer networks, where the existing theories do not seem to apply directly. In this paper, we give a unifying treatment of max-min fairness, which encompasses all existing results in a simplifying framework, and extends its applicability to new examples. First, we observe that the existence of max-min fairness is actually a geometric property of the set of feasible allocations (uniqueness always holds). There exist sets on which max-min fairness does not exist, and we describe a large class of sets on which a max-min fair allocation does exist. This class contains the compact, convex sets of \RRN\RR^N, but not only. Second, we give a general purpose, centralized algorithm, called Max-min Programming, for computing the max-min fair allocation in all cases where it exists (whether the set of feasible allocations is in our class or not). Its complexity is of the order of NN linear programming steps in \RRN\RR^N, in the case where the feasible set is defined by linear constraints. We show that, if the set of feasible allocations has the free-disposal property, then Max-min Programming degenerates to a simpler algorithm, called Water Filling, whose complexity is much less. Free disposal corresponds to the cases where a bottleneck argument can be made, and Water Filling is the general form of all previously known centralized algorithms for such cases. Our derivations are based on the relation between max-min fairness and leximin ordering. All our results apply mutatis mutandis to min-max fairness. Our results apply to weighted, unweighted and util-max-min and min-max fairness. Distributed algorithms for the computation of max-min fair allocations are left outside the scope of this paper.

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