Scheduling Transfers of Resources over Time: Towards Car-Sharing with Flexible Drop-Offs

We consider an offline car-sharing assignment problem with flexible drop-offs, in which n users (customers) present their driving demands, and the system aims to assign the cars, initially located at given locations, to maximize the number of satisfied users. Each driving demand specifies the pick-up location and the drop-off location, as well as the time interval in which the car will be used. If a user requests several driving demands, then she is satisfied only if all her demands are fulfilled. We show that minimizing the number of vehicles that are needed to fulfill all demands is solvable in polynomial time. If every user has exactly one demand, we show that for given number of cars at locations, maximizing the number of satisfied users is also solvable in polynomial time. We then study the problem with two locations a and b, and where every user has two demands: one demand for transfer from a to b, and one demand for transfer from b to a, not necessarily in this order. We show that maximizing the number of satisfied users is np-hard, and even apx-hard, even if all the transfers take exactly the same (non-zero) time. On the other hand, if all the transfers are instantaneous, the problem is again solvable in polynomial time.