# Max Flow = Min Cut

Explore the connection between maximum flows and minimum cuts.

In this lesson, we want to prove that the Ford-Fulkerson method is actually guaranteed to find a maximum flow. To do this, we’ll take a short detour into the territory of another graph problem, finding minimum cuts.

## The minimum cut problem

The minimum cut problem is another problem that can be posed in a flow network $G = (V, E, c)$. Given a source $s$ and a sink $t$, an $\mathbf{s}$**-**$\mathbf{t}$**-cut** is a partition of the vertices $V$ into two subsets $S$ and $T$, such that $s \in S$ and $t \in T$.

The value $v(S, T)$ of an $s$-$t$-cut is the capacity of all edges that cross the cut from $S$ to $T$.

$v(S, T) = \sum_{u \in S, v \in T, (u, v) \in E} c(u, v).$

As an example, let’s take another look at the flow network from the previous lesson:

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