[Paper Review] Improved Approximation Algorithms for the Non-preemptive Speed-scaling Problem

Evripidis Bampis, Giorgio Lucarelli+1 more|arXiv (Cornell University)|Sep 28, 2012

Scheduling and Optimization Algorithms11 references3 citations

TL;DR

This paper presents improved approximation algorithms for the non-preemptive speed-scaling problem, where jobs must execute without interruption. It introduces a novel algorithmic framework that partitions jobs into agreeable, clique, or laminar instances and applies a two-phase approach combining optimal preemptive scheduling with non-preemptive speed adjustment, achieving a $(4(2 - 1/m))^{α-1}$-approximation ratio for agreeable instances on $m$ processors.

ABSTRACT

We are given a set of jobs, each one specified by its release date, its deadline and its processing volume (work), and a single (or a set of) speed-scalable processor(s). We adopt the standard model in speed-scaling in which if a processor runs at speed s then the energy consumption is s^α per time unit, where α>1. Our goal is to find a schedule respecting the release dates and the deadlines of the jobs so that the total energy consumption is minimized. While most previous works have studied the preemptive case of the problem, where a job may be interrupted and resumed later, we focus on the non-preemptive case where once a job starts its execution, it has to continue until its completion without any interruption. We propose improved approximation algorithms for particular instances of the multiprocessor non-preemptive speed-scaling problem.

Motivation & Objective

• Address the non-preemptive speed-scaling problem, where job execution cannot be interrupted, to minimize energy consumption under dynamic voltage/frequency scaling.
• Overcome the strong NP-hardness of the non-preemptive variant by designing efficient approximation algorithms for special instance families.
• Focus on practical and structured input classes such as agreeable, clique, and laminar instances, which model real-world scheduling constraints.
• Provide improved approximation guarantees over prior work, particularly for multiprocessor settings with $m \geq 1$.
• Establish a framework that leverages optimal preemptive schedules as a lower-bound reference to guide non-preemptive scheduling decisions.

Proposed method

• Partition the input jobs into $k$ consecutive time-interval-based sub-instances $\mathcal{I}_1, \dots, \mathcal{I}_k$ based on critical time points $T_1, \dots, T_k$.
• For each sub-instance $\mathcal{I}_\ell$, compute an optimal preemptive schedule $\mathcal{S}_{pr}^{(\ell)}$ using known polynomial-time algorithms.
• Transform each preemptive schedule $\mathcal{S}_{pr}^{(\ell)}$ into a non-preemptive schedule $\mathcal{S}_{npr}^{(\ell)}$ by adjusting job speeds and execution intervals to respect non-preemption constraints.
• Use a key structural property: for any two jobs from adjacent sub-instances, their active intervals do not overlap, ensuring schedule feasibility.
• Apply a speed-scaling transformation that increases job speed by a factor of 2 to compress execution time, increasing energy cost by $2^{\alpha-1}$, which is bounded via Proposition 6.
• Concatenate the non-preemptive schedules $\mathcal{S}_{npr}^{(\ell)}$ to form the final schedule $\mathcal{S}_{npr}$, ensuring all release and deadline constraints are met.

Experimental results

Research questions

• RQ1 Can we design a constant-factor approximation algorithm for the non-preemptive speed-scaling problem on multiple processors?
• RQ2 What is the best achievable approximation ratio for non-preemptive scheduling in agreeable, clique, or laminar instances?
• RQ3 How does the energy cost of a non-preemptive schedule relate to that of an optimal preemptive schedule under the same constraints?
• RQ4 Can we decompose a general instance into sub-instances such that the resulting non-preemptive schedule remains feasible and near-optimal?
• RQ5 What is the impact of time interval compression (via speed-up) on energy consumption in non-preemptive settings?

Key findings

• The proposed algorithm achieves a $(4(2 - 1/m))^{α-1}$-approximation ratio for the non-preemptive speed-scaling problem on $m$ processors for agreeable instances.
• For each sub-instance $\mathcal{I}_\ell$, the non-preemptive schedule $\mathcal{S}_{npr}^{(\ell)}$ is at most $(2(2 - 1/m))^{α-1}$ times more expensive than the optimal preemptive schedule $\mathcal{S}_{pr}^{(\ell)}$.
• The energy cost of the transformed schedule $\mathcal{S}_{npr}^{(\ell)}$ is bounded by $2^{\alpha-1}$ times the cost of a preemptive schedule on a compressed interval, as formalized in Proposition 6.
• The concatenation of non-preemptive schedules across sub-instances remains feasible because the deadline of any job in $\mathcal{I}_\ell$ is strictly before the release time of any job in $\mathcal{I}_{\ell+1}$.
• The total energy of the final non-preemptive schedule $\mathcal{S}_{npr}$ is bounded by $(4(2 - 1/m))^{α-1}$ times the energy of an optimal preemptive schedule for the full instance.
• The framework is general and applies to special cases such as common release dates, common deadlines, and pure-laminar instances, which are subsumed by the agreeable class.