Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Minimizing the Weighted Makespan with Restarts on a Single Machine

Aflatoun Amouzandeh, Klaus Jansen, Lis Pirotton, Rob van Stee, Corinna Wambsganz

TL;DR

The paper investigates online scheduling on a single machine with restart interruptions to minimize the weighted makespan $WC_{\max}$ under release dates. It establishes a general lower bound of $1.4656$ on the competitive ratio for deterministic online algorithms and introduces a unit-sized job algorithm that achieves a competitive ratio better than $1.3098$, complemented by a $1.2344$ lower bound for the unit-sized case. The key methodological contributions include the design and analysis of the Limited Largest Weight (llw) algorithm and detailed competitive analysis that ties performance to roots of cubic equations, reflecting the restart dynamics. These results advance our understanding of restart-enabled online scheduling and provide tight benchmarks and techniques that may inform future improvements for more general processing times.

Abstract

We consider the problem of minimizing the weighted makespan on a single machine with restarts. Restarts are similar to preemptions but weaker: a job can be interrupted, but then it has to be run again from the start instead of resuming at the point of interruption later. The objective is to minimize the weighted makespan, defined as the maximum weighted completion time of jobs. We establish a lower bound of 1.4656 on the competitive ratio achievable by deterministic online algorithms. For the case where all jobs have identical processing times, we design and analyze a deterministic online algorithm that improves the competitive ratio to better than 1.3098. Finally, we prove a lower bound of 1.2344 for this case.

Minimizing the Weighted Makespan with Restarts on a Single Machine

TL;DR

The paper investigates online scheduling on a single machine with restart interruptions to minimize the weighted makespan under release dates. It establishes a general lower bound of on the competitive ratio for deterministic online algorithms and introduces a unit-sized job algorithm that achieves a competitive ratio better than , complemented by a lower bound for the unit-sized case. The key methodological contributions include the design and analysis of the Limited Largest Weight (llw) algorithm and detailed competitive analysis that ties performance to roots of cubic equations, reflecting the restart dynamics. These results advance our understanding of restart-enabled online scheduling and provide tight benchmarks and techniques that may inform future improvements for more general processing times.

Abstract

We consider the problem of minimizing the weighted makespan on a single machine with restarts. Restarts are similar to preemptions but weaker: a job can be interrupted, but then it has to be run again from the start instead of resuming at the point of interruption later. The objective is to minimize the weighted makespan, defined as the maximum weighted completion time of jobs. We establish a lower bound of 1.4656 on the competitive ratio achievable by deterministic online algorithms. For the case where all jobs have identical processing times, we design and analyze a deterministic online algorithm that improves the competitive ratio to better than 1.3098. Finally, we prove a lower bound of 1.2344 for this case.

Paper Structure

This paper contains 3 sections, 4 theorems, 9 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Lower Bound
  3. Unit-Sized Jobs

Key Result

theorem 1.1

Any algorithm for the problem $1|r_j, \text{online} ,\text{restart} | W C_{\max}$ is at least $R_1\approx 1.4656$ competitive which is the real root of the equation $R_1^3-R_1^2-1=0$.

Figures (2)

  • Figure 1: This example shows the behavior of our algorithm.
  • Figure 2: This figure illustrates the set $S$ with jobs 1 up to $k$. Job $k$ is the critical job, and the jobs after $k$ are not in $S$ and are lighter.

Theorems & Definitions (9)

  • theorem 1.1
  • proof
  • lemma thmcounterlemma
  • proof
  • definition thmcounterdefinition
  • proposition thmcounterproposition
  • proof
  • theorem 1.2
  • proof