Car-Sharing on a Star Network: On-Line Scheduling with k Servers

We study an on-line scheduling problem that is motivated by applications such as car-sharing for trips between an airport and a group of hotels. Users submit ride requests, and the scheduler aims to accept requests of maximum total profit using k servers (cars). Each ride request specifies the pick-up time, the pick-up location, and the drop-off location, where one of the two locations must be the airport. A request must be submitted a fixed amount of time before the pick-up time. The scheduler has to decide whether or not to accept a request immediately at the time when the request is submitted (booking time). In the unit travel time variant, the travel time between the airport and any hotel is a fixed value t. We give a 2-competitive algorithm for the case in which the booking interval (pick-up time minus booking time) is at least t and the number of servers is even. In the arbitrary travel time variant, the travel time between the airport and a hotel may have arbitrary length between t and L t for some L >= 1. We give an algorithm with competitive ratio O(log L) if the number of servers is at least ceil[log L]. For both variants, we prove matching lower bounds on the competitive ratio of any deterministic on-line algorithm