Optimal Flow Distribution Among Multiple Channels with Unknown Capacities

Consider a simple network flow problem in which a flow of value D must be split among n channels directed from a source to a sink. The initially unknown channel capacities can be probed by attempting to send a flow of at most D units through the network. If the flow is not feasible, we are told on which channels the capacity was exceeded (binary feedback) and possibly also how many units of flow were successfully sent on these channels (throughput feedback). For throughput feedback we present optimal protocols for minimizing the number of rounds needed to find a feasible flow and for minimizing the total amount of wasted flow. For binary feedback we present an asymptotically optimal protocol.