Online car-sharing problem with variable booking times

In this paper, we address the problem of online car-sharing with variable booking times (CSV for short). In this scenario, customers submit ride requests, each specifying two important time parameters: the booking time and the pick-up time (start time), as well as two location parameters—the pick-up location and the drop-off location within a graph. For each request, it’s important to note that it must be booked before its scheduled start time. The booking time can fall within a specific interval prior to the request’s starting time. Additionally, each car is capable of serving only one request at any given time. The primary objective of the scheduler is to optimize the utilization of k cars to serve as many requests as possible. As requests arrive at their booking times, the scheduler faces an immediate decision: whether to accept or decline the request. This decision must be made promptly upon request submission, precisely at the booking time. We prove that no deterministic online algorithm can achieve a competitive ratio smaller than \(L+1\) even on a special case of a path (where L denotes the ratio between the largest and the smallest request travel time). For general graphs, we give a Greedy Algorithm that achieves \((3L+1)\)-competitive ratio for CSV. We also give a Parted Greedy Algorithm with competitive ratio \((\frac{5}{2}L+10)\) when the number of cars k is no less than \(\frac{5}{4}L+20\); for CSV on a special case of a path, the competitive ratio of Parted Greedy Algorithm is \((2L+10)\) when \(k\ge L+20\).