Consider the Dual LP: $$ \min\sum_{e\in E}c_ey_e$$ s.t.: $$\sum_{e:e\in p}y_e\ge 1\ \ \text{for each}\ \ p\in \mathcal P\\ y_e\ge 0$$
$e$: Represents an edge in $E$
$c_e$: Max flow capacity of an edge $e$
$y_e$: Represents a 0 or 1 (ideally we want all edges in the min-cut to be 1 and everything else 0)
$P$: Is so of all paths from sink to target
$p$: Is a sink to target path in $P$
$E$: Is the set of edges
So I have this Dual LP and I need to convert it to a network flow problem. I don't really know how to because there's no notion of which edges $e$ connect to vertices which then connect to other edges $e$.