Optimal Flow Distribution Among Multiple Channels with Unknown Capacities

Consider a simple network flow problem in which there are n channels directed from a source to a sink. The channel capacities are unknown and we wish to determine a feasible network flow of value D. Flow problems with unknown capacities arise naturally in numerous applications, including inter-domain traffic routing in the Internet, bandwidth allocation for sending files in peer-to-peer networks, but also the distribution of physical goods like newspapers among different points of sale. We study protocols that probe the network by attempting to send a flow of at most D units through the network. If the flow is not feasible, the protocol is told on which channels the capacity was exceeded (binary feedback) and possibly also how many units of flow were successfully send on these channels (throughput feedback). For the latter, more informative type of feedback we present optimal protocols for minimizing the number rounds needed to find a feasible flow and for minimizing the total amount of wasted flow. For binary feedback, we show that one can exploit the fact that network capacities are often larger than the demand D: We present a protocol for this situation that is optimal under certain assumptions and finds solution more quickly than the generalized binary search protocol previously proposed in the literature. For the special case of two channels we present a protocol that is optimal without any extra assumptions and outperforms binary search.