# Why Locally-Fair Maximal Flows in Client-Server Networks Perform Well

### Venue

Computing and Combinatorics, Springer Berlin Heidelberg, 12715 NE 81st PL (2009), pp. 368-377

### Publication Year

2009

### Authors

Chad Yoshikawa, Ken Berman

### BibTeX

## Abstract

Maximal flows reach at least a 1/2 approximation of the maximum flow in
client-server networks. By adding only 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,Δ2−δ2Δ2−δΔ−Δ
}, 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.