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

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 it; in most, but not all cases, they are based on the notion of bottlenecks. In spite of this wide applicability, there are still...

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Bibliographic Details
Published in:IEEE/ACM transactions on networking Vol. 15; no. 5; pp. 1073 - 1083
Main Authors: Radunovic, B., Le Boudec, J.-Y.
Format: Journal Article
Language:English
Published: New York IEEE 01.10.2007
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Algorithms
Allocations
Best-effort traffic
Complexity
Computation
Design optimization
Distributed algorithms
Distributed computing
elastic traffic
Filling
Linear programming
Mathematical programming
mathematical programming/optimization
max-min fairness
Networks
Peer to peer computing
Programming
Proving
Resource management
Studies
system design
Telecommunication traffic
Water
ISSN:1063-6692, 1558-2566
Online Access:Get full text
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Summary: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 it; 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 wireless 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 extend 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. 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, but is not limited to the compact, convex sets of R N . 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 N linear programming steps in R 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 reduces to a simpler algorithm, called water filling, whose complexity is much lower. 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. 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 outside the scope of this paper.
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ISSN:1063-6692
1558-2566
DOI:10.1109/TNET.2007.896231