Online Algorithms for Machine Minimization

In this paper, we consider the online version of the machine minimization problem (introduced by Chuzhoy et al., FOCS 2004), where the goal is to schedule a set of jobs with release times, deadlines, and processing lengths on a minimum number of identical machines. Since the online problem has strong lower bounds if all the job parameters are arbitrary, we focus on jobs with uniform length. Our main result is a complete resolution of the deterministic complexity of this problem by showing that a competitive ratio of $e$ is achievable and optimal, thereby improving upon existing lower and upper bounds of 2.09 and 5.2 respectively. We also give a constant-competitive online algorithm for the case of uniform deadlines (but arbitrary job lengths); to the best of our knowledge, no such algorithm was known previously. Finally, we consider the complimentary problem of throughput maximization where the goal is to maximize the sum of weights of scheduled jobs on a fixed set of identical machines (introduced by Bar-Noy et al. STOC 1999). We give a randomized online algorithm for this problem with a competitive ratio of e/e-1; previous results achieved this bound only for the case of a single machine or in the limit of an infinite number of machines.

💡 Research Summary

The paper studies three closely related online scheduling problems and establishes tight competitive bounds for each.

1. Online Machine Minimization with Uniform Job Lengths
The authors consider the classic online machine minimization problem where each job has a release time, a deadline, and unit processing time. The goal is to schedule all jobs using as few identical machines as possible, and the performance of an online algorithm is measured by its competitive ratio – the worst‑case ratio between the number of machines it uses and the optimal offline number. Prior work gave an upper bound of 5.2 and a lower bound of 2.09, leaving a large gap.

The paper introduces the deterministic “e‑EDF” algorithm. At any time t the algorithm knows all jobs released so far, computes the optimal offline machine count for this prefix, denoted Offline(t), and opens ⌈e·Offline(t)⌉ machines. Jobs are assigned using the Earliest‑Deadline‑First (EDF) rule. The key technical contribution is a proof that this policy never violates any deadline. The proof proceeds by constructing a fractional schedule that satisfies two conditions: (i) fractional completion – each job with deadline ≤ d receives a total processing amount of 1 before d, and (ii) fractional packing – at any time the sum of fractional loads does not exceed e·Offline(t). The authors define for each job j a function f_j(x)=1/(d−x) on a shortened interval