Algorithmica

, Volume 80, Issue 5, pp 1556–1574 | Cite as

Towards Flexible Demands in Online Leasing Problems

  • Shouwei Li
  • Christine Markarian
  • Friedhelm Meyer auf der Heide
Article
  • 38 Downloads
Part of the following topical collections:
  1. Special Issue on Computing and Combinatorics

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.

Keywords

Online algorithms Leasing Infrastructure problems Parking permit problem Deadlines Set cover leasing Facility leasing 

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Computer Science Department, Heinz Nixdorf InstituteUniversity of PaderbornPaderbornGermany

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