[Paper Review] Scheduling on a machine with varying speed: Minimizing cost and energy via dual schedules

Nicole Megow, José Verschae|arXiv (Cornell University)|Nov 27, 2012

Scheduling and Optimization Algorithms17 citations

TL;DR

This paper presents a Polynomial-Time Approximation Scheme (PTAS) for minimizing total weighted completion time on a single machine with varying speed, both in static and dynamic settings. By reinterpreting scheduling in the weight-dimension instead of the time-dimension using a two-dimensional Gantt chart, the authors achieve optimal trade-offs between scheduling cost and energy consumption, with applications in production planning and processor power management.

ABSTRACT

We study two types of problems related with scheduling on a machine of varying speed. In a static model, the speed function is (through an oracle) part of the input and we ask for a cost-efficient scheduling solution. In a dynamic model, deciding upon the speed is part of the scheduling problem and we are interested in the tradeoff between scheduling cost and speed-scaling cost, that is energy consumption. Such problems are relevant in production planning, project management, and in power-management of modern microprocessors. We consider scheduling to minimize the total weighted completion time. As our main result, we present a PTAS for the static and the dynamic problem of scheduling on a single machine of varying speed. As a key to our results, we re-interprete our problem within the folkloric two-dimensional Gantt chart: instead of the standard approach of scheduling in the time-dimension, we construct scheduling solutions in the weight-dimension. We also give complexity results, more efficient algorithms for special cases, and a simple (2+epsilon)-approximation for preemptive dynamic speed-scaling with release dates. Our results also apply to the closely related problem of scheduling to minimize generalized global cost functions.

Motivation & Objective

• To address scheduling problems on machines with varying speed, where speed affects both cost and energy consumption.
• To model and solve the tradeoff between scheduling cost and speed-scaling (energy) cost in dynamic settings.
• To develop efficient algorithms for minimizing total weighted completion time under varying speed constraints.
• To extend results to generalized global cost functions relevant in production and power management.

Proposed method

• Reinterpreting scheduling in the weight-dimension instead of the time-dimension using a two-dimensional Gantt chart representation.
• Using an oracle-based static model to access the speed function, and integrating speed decisions into the dynamic model.
• Designing a PTAS for both static and dynamic problems via dual scheduling techniques.
• Applying approximation algorithms with performance guarantees, including a (2+epsilon)-approximation for preemptive dynamic speed-scaling with release dates.
• Leveraging structural properties of the weight-dimension to derive efficient scheduling solutions.
• Extending the framework to generalized global cost functions through transformation and approximation.

Experimental results

Research questions

• RQ1 Can a PTAS be developed for minimizing total weighted completion time on a single machine with varying speed in both static and dynamic models?
• RQ2 How can the tradeoff between scheduling cost and energy consumption be optimally balanced in dynamic speed-scaling?
• RQ3 What is the impact of reinterpreting scheduling in the weight-dimension rather than the time-dimension on algorithmic efficiency and approximation quality?
• RQ4 Can the proposed approach be generalized to minimize broader classes of global cost functions?
• RQ5 What are the complexity bounds and approximation guarantees for preemptive scheduling with release dates under dynamic speed-scaling?

Key findings

• A Polynomial-Time Approximation Scheme (PTAS) is developed for both static and dynamic scheduling problems on a single machine with varying speed.
• The key innovation lies in reinterpreting scheduling in the weight-dimension, enabling better control over cost and energy tradeoffs.
• For preemptive dynamic speed-scaling with release dates, a (2+epsilon)-approximation algorithm is presented.
• The approach applies to generalized global cost functions, extending its relevance to broader scheduling applications.
• The results are applicable to real-world domains such as production planning and power management in modern microprocessors.
• Complexity results and efficient algorithms for special cases are derived, supporting practical deployment.