Tight Competitive Analyses of Online Car-sharing Problems

The car-sharing problem, proposed by Luo, Erlebach and Xu in 2018, mainly focuses on an online model in which there are two locations: 0 and 1, and $k$ total cars. Each request which specifies its pick-up time and pick-up location (among 0 and 1, and the other is the drop-off location) is released in each stage a fixed amount of time before its specified start (i.e. pick-up) time. The time between the booking (i.e. released) time and the start time is enough to move empty cars between 0 and 1 for relocation if they are not used in that stage. The model, called $k$S2L-F, assumes that requests in each stage arrive sequentially regardless of the same booking time and the decision (accept or reject) must be made immediately. The goal is to accept as many requests as possible. In spite of only two locations, the analysis does not seem easy and the (tight) competitive ratio (CR) is only known to be 2.0 for $k=2$ and 1.5 for a restricted $k$, i.e., a multiple of three. In this paper, we remove all the holes of unknown CR; namely we prove that the CR is $\frac{2k}{k + \lfloor k/3 \rfloor}$ for all $k\geq 2$. Furthermore, if the algorithm can delay its decision until all requests have come in each stage, the CR is improved to roughly 4/3. We can take this advantage even more, precisely we can achieve a CR of $\frac{2+R}{3}$ if the number of requests in each stage is at most $Rk$, $1 \le R \le 2$, where we do not have to know the value of $R$ in advance. Finally we demonstrate that randomization also helps to get (slightly) better CR's.