What is the duality between path cover and flow? Let there be a bipartite directed graph $G=(V,E)$.
Let's say we have a path cover of the graph. In some texts it is said that this path cover &quot;induces&quot; a flow on $G$. What does this mean?
How can we extract a flow from a path cover (and similarly, extract a path cover from a flow)?
Presumably, the flow would send $k$ units of flow along the edge $(u,v)$, where $k$ is the number of paths in the path cover that traverse that edge.
I haven't verified that this will always be guaranteed to be a valid flow for all possible path covers, but it doesn't seem like it on first glance, so more context might be needed to resolve that.