The following is based on my understanding, which may have flaws, so please correct me if I claim something wrong.

When solving VRPs with Branch-and-Price, one can enforce integrality by branching on arcs. When doing that, enforcing branching on arcs can be done either by adding constraints in the restricted master problem, or by removing infeasible columns and remove arc(s) from the graph used when solving the pricing problem.

In my Branch-and-Price implementation, I have used the former approach to enforce branching on arcs. I know that by doing that I miss a potential speedup that can be gained by reducing the number of arcs in the subproblem graph. Someone told me that I can remove arcs from the graph, even if I enforce branching by adding constraints to the restricted master problem. I find that a bit strange, since the dual variables from the added constraints then do not matter.

So my question is: can you reduce the graph be removing arcs if branching on arcs is enforced by adding constraints to the reduced master problem?

Additionally, what are the pros and cons of the two different approaches to enforce branching on arcs?

The following is based on my understanding, which may have flaws, so please correct me if I claim something wrong.

When solving VRPs with Branch-and-Price, one can enforce integrality by branching on arcs. When doing that, enforcing branching on arcs can be done either by adding constraints in the restricted master problem, or by removing infeasible columns and remove arc(s) from the graph used when solving the pricing problem.

In my Branch-and-Price implementation, I have used the former approach to enforce branching on arcs. I know that by doing that I miss a potential speedup that can be gained by reducing the number of arcs in the subproblem graph. Someone told me that I can remove arcs from the graph, even if I enforce branching by adding constraints to the restricted master problem. I find that a bit strange, since the dual variables from the added constraints then do not matter.

So my question is: can you reduce the graph by removing arcs if branching on arcs is enforced by adding constraints to the reduced master problem?

Additionally, what are the pros and cons of the two different approaches to enforce branching on arcs?