Can anybody describe what the minimum flow in a graph network is please? I am not referring to the Minimum Cost Flow Problem here.
The minimum flow I believe is the opposite to the max flow of a network. The max flow seems intuitive in that you are trying to find the max flow through a network and you could use the min-cut approach.
But I cannot understand the minimum flow of a network. This is a paper that discuss the concept. I thought that the min flow of a network is 0 but obvious that has no value.
For a minimum flow problem, we take a network where each edge $e$ has an interval $[l_e, u_e]$ associated with it; the requirement for a feasible flow is (in addition to the flow condition) that the flow along $e$ must fall into this interval.
This is is a generalization of the networks usually used in the maximum flow problems, where the interval for an edge $e$ with capacity $c_e$ is $[0, c_e]$. However, if $l_e > 0$ for some edges, then the $0$ flow is not feasible, and so the minimum flow is not trivial to find.
Typically, minimum flow algorithms work in two steps: first, we find some feasible flow, and then we try to improve it to be minimum. The first step can be done by a reduction to the maximum flow problem (it's outlined here starting from p. 9, for example). The second step uses specialized algorithms.