Refining the Complexity Landscape of Speed Scaling: Hardness and Algorithms

2025-12-19

AI Summary

This paper investigates the computational complexity of jointly optimizing weighted flow time and energy consumption in speed-scaling single-processor scheduling, focusing on four long-standing open variants. Using constructive reductions, structural analysis of feasible schedules, and dynamic programming, we establish— for the first time—NP-hardness for two classical variants: (i) unit-weight jobs with arbitrary processing times, and (ii) arbitrary-weight jobs with unit processing times, under the flow-time-plus-energy objective. Crucially, we discover that ordering jobs by completion time enables polynomial-time exact optimization, revealing a fundamental departure from conventional priority-based rules. Our work fully characterizes the complexity landscape of all four variants: we prove two tight NP-hardness results and provide exact polynomial-time algorithms based on completion-time ordering. This resolves open problems explicitly posed in prior literature.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments

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.

Problem

• Determines NP-hardness for minimizing flow time plus energy
• Solves open complexity questions for four scheduling variants
• Establishes polynomial solvability with given completion-time ordering

Innovation

• NP-hardness proof for weighted flow time plus energy
• Polynomial-time solution with completion-time ordering
• Complexity settled for four open scheduling variants