The maximum flow problem

In this network we have two special vertices: a source vertex s and a sink vertex t. Our goal is to send the maximum amount of flow from s to t; flow can only travel along arcs in the right direction, and is constrained by the arc capacities. This “flow” could be many things: imagine sending water along pipes, with the capacity representing the size of the pipe; or traffic, with the capacity being the number of lanes of a road. Or a communications network, where the capacity represents the bandwidth of a particular link in the network, and the flow consists of data packets being sent. Network flows appear in many many settings. The key property of a flow is flow conservation. Consider some vertex i, different from the source or the sink. Then the amount of flow going into vertex i should be the same as the amount of flow going out; no flow appears or disappears at this vertex. In the above example, there is a flow of value 9 (you should be able to find such a flow pretty easily). But there is no flow of any larger value. How can we see that this is the case? Well, consider the set S shown in red on the figure. Since s ∈ S and t / ∈ S, all flow from s to t needs to cross the boundary of S. In the example, the total amount of outgoing capacity crossing this boundary is 9; and so there cannot be a flow of larger value. Let’s formalize this notion a bit.