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Towards Flexible Demands in Online Leasing Problems

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Abstract

We consider online leasing problems in which demands arrive over time and need to be served by leasing resources. We introduce a new model for these problems in which a resource can be leased for K different durations each incurring a different cost (longer leases cost less per time unit). Each demand i can be served any time between its arrival \(a_i\) and its deadline \(a_i + d_i\) by a leased resource. The objective is to meet all deadlines while minimizing the total leasing costs. This model is a natural generalization of Meyerson’s ParkingPermitProblem (in: Proceedings of the 46th annual IEEE symposium on foundations of computer science, FOCS ’05, IEEE Computer Society, Washington, pp 274–284, 2005) in which \(d_i=0\) for all i. We propose an online algorithm that is \(\varTheta (K + \frac{d_\textit{max}}{l_\textit{min}})\)-competitive, where \(d_\textit{max}\) and \(l_\textit{min}\) denote the largest \(d_i\) and the shortest available lease length, respectively. We also extend SetCoverLeasing and FacilityLeasing to their respective variants in which deadlines are added. For the former, we give an \(\mathcal {O}\left( \log (m \cdot (K + \frac{d_\textit{max}}{l_\textit{min}}))\log l_\textit{max} \right) \)-competitive randomized algorithm, where m represents the number of subsets and \(l_\textit{max}\) represents the largest available lease length. This improves on existing solutions for the original SetCoverLeasing problem. For the latter, we give an \(\mathcal {O}\left( (K + \frac{d_\textit{max}}{l_\textit{min}})\log l_{\text {max}} \right) \)-competitive deterministic algorithm.

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Authors and Affiliations

  1. Computer Science Department, Heinz Nixdorf Institute, University of Paderborn, Fürstenallee 11, 33102, Paderborn, Germany

    Shouwei Li, Christine Markarian & Friedhelm Meyer auf der Heide

Authors
  1. Shouwei Li
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  2. Christine Markarian
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  3. Friedhelm Meyer auf der Heide
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Corresponding author

Correspondence to Christine Markarian.

Additional information

The results presented in this paper are based on the conference paper [28]. This work was partially supported by the German Research Foundation (DFG) within the Collaborative Research Center ‘On-The-Fly Computing’ (SFB 901).

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Li, S., Markarian, C. & Meyer auf der Heide, F. Towards Flexible Demands in Online Leasing Problems. Algorithmica 80, 1556–1574 (2018). https://doi.org/10.1007/s00453-018-0420-y

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