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Tight Competitive Analyses of Online Car-Sharing Problems

Authors Ya-Chun Liang, Kuan-Yun Lai, Ho-Lin Chen, Kazuo Iwama



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Author Details

Ya-Chun Liang
  • Department of Industrial Engineering and Engineering Management, National Tsing Hua University, Hsinchu, Taiwan
Kuan-Yun Lai
  • Department of Industrial Engineering and Engineering Management, National Tsing Hua University, Hsinchu, Taiwan
Ho-Lin Chen
  • Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan
Kazuo Iwama
  • Department of Industrial Engineering and Engineering Management, National Tsing Hua University, Hsinchu, Taiwan

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Ya-Chun Liang, Kuan-Yun Lai, Ho-Lin Chen, and Kazuo Iwama. Tight Competitive Analyses of Online Car-Sharing Problems. In 32nd International Symposium on Algorithms and Computation (ISAAC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 212, pp. 50:1-50:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.ISAAC.2021.50

Abstract

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 kS2L-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 value of k, i.e., a multiple of three. In this paper, we remove all the holes of unknown CR’s; namely we prove that the CR is 2k/(k + ⌊k/3⌋) for all k ≥ 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 further, precisely we can achieve a CR of (2+R)/3 if the number of requests in each stage is at most Rk, 1 ≤ R ≤ 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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Car-sharing
  • Competitive analysis
  • On-line scheduling
  • Randomized algorithm

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References

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