I am solving a minimization problem with a column generation procedure. The master problem is of the form $$ \min \sum_{i\in \Omega}c_i \lambda_i $$ subject to $$ \sum_{i\in \Omega \mid v \in i } \lambda_i = 1 \quad \forall v \in V $$ Columns $\lambda_i$ represent paths with nodes belonging to set $V$.
Let $\pi_v$ denote the dual variable associated with each constraint. In my subproblem, I am looking for columns that have smallest marginal cost : $$ \hat{c}_i = c_i - \sum_{v\in V, v \in i} \pi_v $$
My question is, if $\pi_v <0$, can I safely delete node $v$ from the subproblem (as passing through $v$ will increase the global cost) ? Or do I have to keep it, in case passing through $v$ is the only way to access other nodes that together have a potential negative reduced cost ?
More generally, is there a strategy to delete useless nodes or edges from the subproblem to speed up computation ?
In this article by Francois Soumis et al (a VRP variant solved with column generation), edges and nodes are eliminated heuristically from the sunproblem: [p.15, section 4.3]
They refer the reader to another paper ([14]) for the details of these "heuristics," but the paper is nowhere to be found. I am interested in such techniques as my problem is also a VRP variant (my subproblem also consists in finding some sort of shortest path).