Improved Algorithms for Online Rent Minimization Problem Under Unit-Size Jobs

TL;DR

This paper improves the competitive ratio for online rent minimization with unit-size jobs from approximately 15.16 to 6, using a new oracle-based online algorithm framework.

Contribution

It introduces a novel oracle-based online algorithm that significantly reduces the competitive ratio for the problem under unit-size jobs.

Findings

Achieved a competitive ratio of 6 for online rent minimization with unit jobs.

Improved upon the previous best ratio of approximately 15.16.

Demonstrated effectiveness of oracle-based algorithms in online scheduling.

Abstract

We consider the Online Rent Minimization problem, where online jobs with release times, deadlines, and processing times must be scheduled on machines that can be rented for a fixed length period of $T$. The objective is to minimize the number of machine rents. This problem generalizes the Online Machine Minimization problem where machines can be rented for an infinite period, and both problems have an asymptotically optimal competitive ratio of $O(\log(p_{\max}/p_{\min}))$ for general processing times, where $p_{\max}$ and $p_{\min}$ are the maximum and minimum processing times respectively. However, for small values of $p_{\max}/p_{\min}$, a better competitive ratio can be achieved by assuming unit-size jobs. Under this assumption, Devanur et al. (2014) gave an optimal $e$-competitive algorithm for Online Machine Minimization, and Chen and Zhang (2022) gave a $(3e+7)\approx 15.16$-competitive algorithm for Online Rent Minimization. In this paper, we significantly improve the competitive ratio of the Online Rent Minimization problem under unit size to $6$, by using a clean oracle-based online algorithm framework.