Single-vehicle scheduling with time window constraints

In the single-vehicle scheduling problem with time window constraints, a vehicle has to visit a set of sites on a graph, and each site must be visited after its ready time but no later than its deadline. The goal is to minimize the total time taken to visit all sites. We prove the conjecture proposed by Psaraftis et al.: if the topological graph is a straight line, the problems are NP-hard for both part and tour version. In addition, we give an O(n²) algorithm to solve a special case where all sites have a common ready time. This algorithm illustrates a duality relationship between the vehicle scheduling problems with arbitrary ready times and that with arbitrary deadlines on a straight line.