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Online Makespan Minimization: Beat LPT by Dynamic Locking

Zhaozi Wang, Zhiwei Ying, Yuhao Zhang

TL;DR

The paper advances online makespan minimization with release times by introducing Generalized SLEEPY, a dynamic-locking extension of SLEEPY that preserves LPT's structure while selectively locking machines. Through a careful combination of bin-packing and efficiency arguments, the authors prove a $1.5-\frac{1}{O(m^2)}$ competitive ratio for fixed $m\ge4$, and show a tight $1.482$ bound for $m=3$ without dynamic locking; they also establish a hardness result showing the need for dynamic locking to beat $1.5$ for large $m$. Central to the analysis is a Left-over Lemma-based framework bounding OPT versus the online algorithm via quantified waste and extended processing, plus a detailed structural study of the last scheduling period, locking chains, and critical job pairs (early vs late). The results close a long-standing gap on whether LPT can be decisively surpassed for constant $m$, and introduce dynamic locking as a robust technique for improving online makespan performance with release times.

Abstract

Online makespan minimization is a fundamental problem in scheduling. In this paper, we investigate its over-time formulation, where each job has a release time and a processing time. A job becomes known only at its release time and must be scheduled on a machine thereafter. The Longest Processing Time First (LPT) algorithm, established by Chen and Vestjens (1997), achieves a competitive ratio of $1.5$. For the special case of two machines, Noga and Seiden introduced the SLEEPY algorithm, which achieves a tight competitive ratio of $1.382$. However, for $m \geq 3$, no known algorithm has convincingly surpassed the long-standing $1.5$ barrier. We propose a natural generalization of SLEEPY and show this simple approach can beat the $1.5$ barrier and achieve $1.482$-competitive when $m=3$. However, when $m$ becomes large, we prove this simple generalization fails to beat $1.5$. Meanwhile, we introduce a novel technique called dynamic locking to overcome this new challenge. As a result, we achieve a competitive ratio of $1.5-\frac{1}{O(m^2)}$, which beats the LPT algorithm ($1.5$-competitive) for every constant $m$.

Online Makespan Minimization: Beat LPT by Dynamic Locking

TL;DR

The paper advances online makespan minimization with release times by introducing Generalized SLEEPY, a dynamic-locking extension of SLEEPY that preserves LPT's structure while selectively locking machines. Through a careful combination of bin-packing and efficiency arguments, the authors prove a competitive ratio for fixed , and show a tight bound for without dynamic locking; they also establish a hardness result showing the need for dynamic locking to beat for large . Central to the analysis is a Left-over Lemma-based framework bounding OPT versus the online algorithm via quantified waste and extended processing, plus a detailed structural study of the last scheduling period, locking chains, and critical job pairs (early vs late). The results close a long-standing gap on whether LPT can be decisively surpassed for constant , and introduce dynamic locking as a robust technique for improving online makespan performance with release times.

Abstract

Online makespan minimization is a fundamental problem in scheduling. In this paper, we investigate its over-time formulation, where each job has a release time and a processing time. A job becomes known only at its release time and must be scheduled on a machine thereafter. The Longest Processing Time First (LPT) algorithm, established by Chen and Vestjens (1997), achieves a competitive ratio of . For the special case of two machines, Noga and Seiden introduced the SLEEPY algorithm, which achieves a tight competitive ratio of . However, for , no known algorithm has convincingly surpassed the long-standing barrier. We propose a natural generalization of SLEEPY and show this simple approach can beat the barrier and achieve -competitive when . However, when becomes large, we prove this simple generalization fails to beat . Meanwhile, we introduce a novel technique called dynamic locking to overcome this new challenge. As a result, we achieve a competitive ratio of , which beats the LPT algorithm (-competitive) for every constant .
Paper Structure (13 sections, 22 theorems, 41 equations, 2 figures, 1 table)

This paper contains 13 sections, 22 theorems, 41 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our Techniques
  3. Paper Organization
  4. Further Related Work
  5. Preliminaries
  6. The Generalized SLEEPY Algorithm
  7. Algorithm's Efficiency
  8. Overview of Our Analysis
  9. Basic Discussion: Last Scheduling Period
  10. Further Discussion of Critical Job Pairs
  11. Early Critical Pairs Only
  12. Late Critical Pair: $\min \{s^\pi_a,s^\pi_b\} < \theta^+$
  13. Late Critical Pair: $\min \{s^\pi_a,s^\pi_b\} \ge \theta^+$

Key Result

theorem thmcountertheorem

Generalized SLEEPY achieves a competitive ratio of $1.5-\frac{1}{O(m^2)}$ for the case $m\geq 4$, with dynamic locking.

Figures (2)

  • Figure 1: Locking chain (the white jobs) and critical jobs (the white and gray jobs).
  • Figure 2: Locking chain (the white jobs) and critical jobs (the white and gray jobs).

Theorems & Definitions (45)

  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • theorem thmcountertheorem
  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • proof
  • lemma thmcounterlemma: Left-over Lemma DBLP:conf/approx/0002KTWZ18
  • lemma thmcounterlemma
  • proof
  • ...and 35 more