Feasibility of the Pickup and Delivery Problem with Fixed Partial Routes: A Complexity Analysis

In the pickup and delivery problem (PDP) a fleet of vehicles must serve customers' requests, which consist of transporting objects from their origins to their destinations. We introduce the PDP with fixed partial routes (PDP-FPR), in which some partial routes are given, and the problem consists in obtaining a solution (a set of routes) that includes those partial routes. We have analyzed the complexity of determining whether or not a feasible solution exists for this problem as well as for some relaxations of it. Checking the feasibility of the PDP-FPR and some of its relaxations is shown to be NP-complete, whereas for other relaxations, the problem was proved to be polynomial-time solvable.