My opinion is: one might not have to stick at branch and price (or, a column generation algorithm).
There are 2 iterative procedures regarding large-scale LP/MIP:
- iteratively adding cutting planes, aka outer approximation, constraint generation
- iteratively recruiting vertices of the original polyhedron, aka inner approximation, column generation
They essentially have a dual relationship—meaning that you only have to opt one of the two methods. The outer approximation cutting plane method is more widespread and technologically mature (e.g. the Gurobi solver is a branch-and-cut solver), thereby more advisable.
here is an example that might be illuminating.