I need help with the following practice problem on network flow:
A cohort of $k$ spies resident in a certain country needs escape routes in case of emergency. They will be travelling using the railway system which we can think of as a directed graph $G = (V, E)$ with $V$ being the cities. Each spy $i$ has a starting point $s_i\in V$ and needs to reach the consulate of a friendly nation; these consulates are in a known set of cities $T\subseteq V$. In order to move undetected, the spies agree that at most $c$ of them should ever pass through any one city. Our goal is to find a set of paths for each of the spies (or detect that the requirements cannot be met). Model this problem as a flow network. Specify the vertices, edges and capacities, and show that a maximum flow in your network can be transformed into an optimal solution for the original problem. You do not need to explain how to solve the max-flow instance itself.
My thought was to make all edge weights have capacity $c$. Then, just use Ford-Fulkerson to find paths in the residual graph. But I'm not sure if this is the right approach because maybe we will have more than $c$ go through a node.