Towards Flexible Demands in Online Leasing Problems

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.