Refining the Complexity Landscape of Speed Scaling: Hardness and Algorithms

TL;DR

This paper investigates the computational complexity of scheduling jobs on speed-scalable processors, establishing NP-hardness for certain variants and polynomial solvability when a completion order is provided, thus clarifying the problem landscape.

Contribution

It resolves the open complexity status of four key scheduling variants, showing NP-hardness in some cases and polynomial-time solvability under specific orderings.

Findings

NP-hard for unit-weight jobs with arbitrary sizes

NP-hard for arbitrary-weight jobs with unit sizes

Polynomial-time solvable with a given completion order

Abstract

We study the computational complexity of scheduling jobs on a single speed-scalable processor with the objective of capturing the trade-off between the (weighted) flow time and the energy consumption. This trade-off has been extensively explored in the literature through a number of problem formulations that differ in the specific job characteristics and the precise objective function. Nevertheless, the computational complexity of four important problem variants has remained unresolved and was explicitly identified as an open question in prior work. In this paper, we settle the complexity of these variants. More specifically, we prove that the problem of minimizing the objective of total (weighted) flow time plus energy is NP-hard for the cases of (i) unit-weight jobs with arbitrary sizes, and (ii)~arbitrary-weight jobs with unit sizes. These results extend to the objective of minimizing the total (weighted) flow time subject to an energy budget and hold even when the schedule is required to adhere to a given priority ordering. In contrast, we show that when a completion-time ordering is provided, the same problem variants become polynomial-time solvable. The latter result highlights the subtle differences between priority and completion orderings for the problem.