Shortest Path Based Column Generation on Large Networks with Many Resource Constraints

We present a relaxation-based dynamic programming algorithm for solving resource constrained shortest path problems arising in column generation pricing problems for formulations involving extremely large networks and a huge number of local resource constraints. Such formulations typically occur when time-expanded networks are used to model a problem. The relaxation-based dynamic programming algorithm alternates between a forward and a backward search. Each search employs bounds derived in the previous search to prune the search space. Between consecutive searches, the relaxation is tightened using a set of critical resources and a set of critical arcs over which these resources are consumed. As a result, a relatively small state space is maintained and many paths can be pruned while guaranteeing that an optimal path is ultimately found.