Refining the Complexity Landscape of Speed Scaling: Hardness and Algorithms

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.

💡 Research Summary

This paper investigates the computational complexity of scheduling jobs on a single speed‑scalable processor while balancing two competing objectives: (weighted) flow time and energy consumption. The authors focus on four problem variants that have remained open in the literature: FE‑IDUA, FE‑ICUA, FE‑IDWU, and FE‑ICWU, where the objective is to minimize the sum of flow time and energy (or, alternatively, to minimize flow time subject to an energy budget). The notation ★‑I/F‑C/D‑W/U‑A/U encodes, respectively, whether the objective is a flow‑plus‑energy (FE) or a budget (B) formulation, whether flow is integral, whether speeds are continuous or discrete, whether jobs are weighted, and whether job sizes are arbitrary or unit.

The first major contribution is a set of NP‑hardness proofs for all four variants. For the unit‑weight, arbitrary‑size case (FE‑IDUA) the authors reduce from the known NP‑hard two‑speed budget problem B‑IDUA. They construct an instance that adds a large‑volume “big” job together with many small jobs released far later. In any optimal schedule the small jobs must be processed immediately to keep flow time low, forcing the big job to finish before them and thereby simulating the original energy‑budget constraint within the flow‑plus‑energy objective. For the arbitrary‑weight, unit‑size case (FE‑IDWU) they give a direct reduction from Subset‑Sum. By introducing a large number of low‑weight unit‑size jobs and carefully setting their weights, they create a gap between YES and NO instances that forces any feasible schedule to either exceed the energy budget or incur a large flow‑time penalty. Both reductions are extended to the continuous‑speed model, establishing that the hardness holds regardless of whether speeds are discrete or continuous.

The second major contribution concerns the role of information about job ordering. The authors distinguish between a priority ordering (which job should be processed next among those released) and a completion‑time ordering (the exact order in which jobs finish). They show that even when a priority ordering is prescribed, the problem remains NP‑hard because determining the optimal speed profile is still entangled with the order of execution. In contrast, when a completion‑time ordering is given, the problem simplifies dramatically: the only remaining decision is the speed at which each job should be run. Leveraging this, they formulate a linear program (LP) for every ★‑ID★‑C variant. The LP variables represent the amount of time each job spends at each allowable speed; constraints enforce that the total processed volume matches each job’s requirement, that the prescribed completion order is respected, and, for budget versions, that total energy does not exceed the budget. The objective is linear (either total flow time or flow plus energy). Because the LP is polynomial‑size and solvable in polynomial time, they obtain a polynomial‑time algorithm for all variants when a completion‑time ordering is supplied. The same LP framework adapts to the budget versions with a minor modification of the objective and an additional energy‑budget constraint.

Additional insights include: (i) a counter‑example to a previously claimed “straightforward” extension of a two‑speed algorithm to k speeds, showing that the extension fails; (ii) an LP‑based algorithm that works for any number of speeds; (iii) a discussion of how the hardness reductions can be extended to enforce a given priority ordering, reinforcing that the difficulty stems from the simultaneous choice of order and speed.

In summary, the paper completes the complexity landscape for flow‑time plus energy scheduling on a single speed‑scalable processor: it proves NP‑hardness for the four previously unresolved variants, demonstrates that prescribing a completion‑time order collapses the problem to polynomial time via an LP formulation, and clarifies the subtle distinction between priority and completion orderings. These results deepen our understanding of where the computational difficulty lies in energy‑aware scheduling and provide a clear direction for future algorithmic work, especially in settings where partial ordering information is available.