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

Maximal flows reach at least a 1/2 approximation of the maximum flow in client-server networks. In this paper we show that by adding only 1 additional time round to any distributed maximal flow algorithm 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, Delta = max (d(u) | u in U) and delta = min (d(v) | v in V). We show that a locally-fair maximal flow f achieves an approximation to the maximum flow of min (1, (Delta^2 - delta)/(2 Delta^2 - delta*Delta - Delta) ) and this result is sharp for any given integers delta and Delta. 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) inreal client-server systems.

• Algorithms and theory