the value of a flow

If a network is a system of pipes from a pump to a drain, the value of a flow is the single number you actually care about: how much water per second is genuinely getting from the pump to the drain. Not how much sloshes around inside, but the net throughput from source to sink. Maximizing this number is the whole game.

Formally, the value of a flow f, written |f|, is the net flow out of the source: the total flow on edges leaving s minus the total flow on edges entering s. A clean fact, which conservation guarantees, is that this also equals the net flow into the sink t — nothing is lost or created in between, so whatever leaves the source must arrive at the sink. You can see why with a balance argument: sum the conservation equation over all vertices except s and t; everything cancels internally, leaving (out of s) minus (into s) equal to (into t) minus (out of t). A second clean fact: for ANY s-t cut, the value of f equals the net flow crossing that cut, which is at most the cut's capacity — so every cut gives an upper bound on |f|.

The value is the objective of the maximum flow problem: find a legal flow with the largest possible |f|. It is also the quantity that the max-flow min-cut theorem pins down exactly, equating the maximum value to the minimum cut capacity. A common slip is to confuse the value with the total amount of flow sitting on all edges; a flow that loops 100 units around a useless cycle can have value 0. Only the net source-to-sink throughput counts.