# Maximizing Online Utilization with Commitment

## TL;DR

This paper develops and analyzes online algorithms for scheduling jobs on parallel machines with commitment constraints, aiming to maximize total accepted processing time, and demonstrates their competitive ratios and improvements over greedy policies.

## Contribution

It introduces new deterministic online algorithms with near-optimal competitive ratios for both preemptive and non-preemptive cases in parallel machine scheduling with commitment.

## Key findings

- Deterministic preemptive algorithm achieves near-tight competitive ratio for any number of machines.
- Non-preemptive algorithm guarantees competitive ratio close to the optimal for single machine.
- Competitive ratios improve exponentially with the number of machines, especially for small epsilon.

## Abstract

We investigate online scheduling with commitment for parallel identical machines. Our objective is to maximize the total processing time of accepted jobs. As soon as a job has been submitted, the commitment constraint forces us to decide immediately whether we accept or reject the job. Upon acceptance of a job, we must complete it before its deadline $d$ that satisfies $d \geq (1+\epsilon)\cdot p + r$, with $p$ and $r$ being the processing time and the submission time of the job, respectively while $\epsilon>0$ is the slack of the system. Since the hard case typically arises for near-tight deadlines, we consider $\varepsilon\leq 1$. We use competitive analysis to evaluate our algorithms. Our first main contribution is a deterministic preemptive online algorithm with an almost tight competitive ratio on any number of machines. For a single machine, the competitive factor matches the optimal bound $\frac{1+\epsilon}{\epsilon}$ of the greedy acceptance policy. Then the competitive ratio improves with an increasing number of machines and approaches $(1+\epsilon)\cdot\ln \frac{1+\epsilon}{\epsilon}$ as the number of machines converges to infinity. This is an exponential improvement over the greedy acceptance policy for small $\epsilon$. In the non-preemptive case, we present a deterministic algorithm on $m$ machines with a competitive ratio of $1+m\cdot \left(\frac{1+\epsilon}{\epsilon}\right)^{\frac{1}{m}}$. This matches the optimal bound of $2+\frac{1}{\epsilon}$ of the greedy acceptance policy for a single machine while it again guarantees an exponential improvement over the greedy acceptance policy for small $\epsilon$ and large $m$. In addition, we determine an almost tight lower bound that approaches $m\cdot \left(\frac{1}{\epsilon}\right)^{\frac{1}{m}}$ for large $m$ and small $\epsilon$.