What does “lower bound” mean in circulation problems?

Question . Circulation problems make it possible to have both a lower and an upper boundary of the flow through a certain arc. The upper bound I understand (like pipes, there are only so many things that can go through). However, it’s hard for me to understand the idea of ​​a lower bound. What does it mean? Will there be an algorithm to solve the problem ...

• try to make sure that each arc with a lower boundary will receive at least such a stream, will not work completely if it cannot find a way?
• just ignore the arc if the lower bound cannot be satisfied? This will make more sense to me, but it will mean that there can be arcs with stream 0 in the resulting graph, i.e.

Context . I am trying to find a way to quickly plan a set of events, each of which has a length and a set of possible times for which they can be scheduled. I am trying to reduce this problem to a circulation problem for which efficient algorithms exist.

I put each event in a directed graph as a node and put it with the number of time intervals that it should fill. Then I add all possible moments as nodes, and finally, all time intervals like this (all arcs are on the right):

The first two events have one possible time and length 1, and the last event has a length of 4 and two possible times.

Does this schedule make sense? In particular, will the number of time intervals that are filled be 2 (only “light”) or six, for example, in the picture?

( push-relabel LEMON, .)