New Results on Online Resource Minimization

TL;DR

This paper introduces new algorithms and bounds for online resource minimization, especially for jobs with agreeable deadlines, achieving constant competitive ratios in various settings and improving previous bounds.

Contribution

The paper presents the first constant ratio competitive algorithm for non-preemptive scheduling with agreeable deadlines and improves bounds for preemptive cases, including an O(log n)-competitive algorithm.

Findings

First constant ratio algorithm for non-preemptive agreeable deadlines

LLF algorithm is constant for agreeable jobs in preemptive setting

An O(log n)-competitive algorithm for general preemptive resource minimization

Abstract

We consider the online resource minimization problem in which jobs with hard deadlines arrive online over time at their release dates. The task is to determine a feasible schedule on a minimum number of machines. We rigorously study this problem and derive various algorithms with small constant competitive ratios for interesting restricted problem variants. As the most important special case, we consider scheduling jobs with agreeable deadlines. We provide the first constant ratio competitive algorithm for the non-preemptive setting, which is of particular interest with regard to the known strong lower bound of n for the general problem. For the preemptive setting, we show that the natural algorithm LLF achieves a constant ratio for agreeable jobs, while for general jobs it has a lower bound of Omega(n^(1/3)). We also give an O(log n)-competitive algorithm for the general preemptive problem, which improves upon the known O(p_max/p_min)-competitive algorithm. Our algorithm maintains a dynamic partition of the job set into loose and tight jobs and schedules each (temporal) subset individually on separate sets of machines. The key is a characterization of how the decrease in the relative laxity of jobs influences the optimum number of machines. To achieve this we derive a compact expression of the optimum value, which might be of independent interest. We complement the general algorithmic result by showing lower bounds that rule out that other known algorithms may yield a similar performance guarantee.