Why locally-fair maximal flows in client-server networks perform well

摘要

Maximal flows reach at least a 1/2 approximation of the maximum flow in client-server networks. By adding 1 additional time round to any distributed maximal flow algorithm we show how this 1/2-approximation can be improved on bounded-degree networks. We call these modified maximal flows ‘locally fair’ since there is a measure of fairness prescribed to each client and server in the network. Let N=(U,V,E,b) represent a client-server network with clients U, servers V, network links E, and node capacities b, where we assume that each capacity is at least one unit. Let d(u) denote the b-weighted degree of any node u∈U∪V, Δ=max {d(u)|u∈U} and δ=min {d(v)|v∈V}. We show that a locally-fair maximal flow f achieves an approximation to the maximum flow of min{1,fracvarDelta2-d2varDelta2-dvarDelta-varDeltamin{1,frac{varDelta^{2}-delta}{2varDelta^{2}-deltavarDelta-varDelta} }, and this result is sharp for any given integers δ and Δ. This results are of practical importance since local-fairness loosely models the steady-state behavior of TCP/IP and these types of degree-bounds often occur naturally (or are easy to enforce) in real client-server systems.