Refining the Complexity Landscape of Speed Scaling: Hardness and Algorithms

TL;DR

This work analyzes scheduling on a single speed-scalable processor to minimize the trade-off between energy and total weighted flow time, formalizing FE (flow plus energy) and budget variants within a broad quintuple notation. It resolves the complexity of four open variants by proving NP-hardness for FE-IDUA and FE-IDWU, even under fixed priority ordering, and demonstrates that providing a fixed completion-time ordering yields a polynomial-time solution via a novel LP formulation (and an adaptable budget version). Additionally, the authors develop a combinatorial algorithm for FE-IDUU with unit sizes/weights, based on a kappa-Delta rule, highlighting a tractable path under FIFO-like completion orders. Collectively, these results complete the complexity landscape for flow-time and energy-aware scheduling in speed-scaling models and connect hardness results with practical LP-based and combinatorial algorithms.

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.

Figures (3)

• Figure 1: An instance of B-IDUA (depicted at the top) is transformed into an instance of FE-IDUA (depicted at the bottom).
• Figure 2: A comparison between speeding up job $n+1$ and job $n$. The dashed lines indicate the completion times and show that speeding up job $n+1$ improves the flow of at most $n+1$ jobs, while speeding up job $n$ improves the flow of at least $n+2$ jobs.
• Figure 3: Jobs $n+1,\dots,2n+1$ are added to an optimal schedule for $\mathcal{I}^B$ and run at speed $s_1$. This gives a schedule for $\mathcal{I}^{FE}$ where jobs $1,\dots,n$ have a total processing time of $V-Y$, and are the only jobs that run at a speed higher than $s_1$.

Theorems & Definitions (40)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 6: Affection
• Definition 7
• Definition 8: Shrinking/expanding energy
• Lemma 8
• Corollary 9
• Lemma 11
• Lemma 12