Energy Aware Scheduling for Weighted Completion Time and Weighted Tardiness

The ever increasing adoption of mobile devices with limited energy storage capacity, on the one hand, and more awareness of the environmental impact of massive data centres and server pools, on the other hand, have both led to an increased interest in energy management algorithms. The main contribution of this paper is to present several new constant factor approximation algorithms for energy aware scheduling problems where the objective is to minimize weighted completion time plus the cost of the energy consumed, in the one machine non-preemptive setting, while allowing release dates and deadlines.Unlike previous known algorithms these new algorithms can handle general job-dependent energy cost functions, extending the application of these algorithms to settings outside the typical CPU-energy one. These new settings include problems where in addition, or instead, of energy costs we also have maintenance costs, wear and tear, replacement costs, etc., which in general depend on the speed at which the machine runs but also depend on the types of jobs processed. Our algorithms also extend to approximating weighted tardiness plus energy cost, an inherently more difficult problem that has not been addressed in the literature.

💡 Research Summary

The paper addresses the increasingly important problem of jointly minimizing scheduling quality‑of‑service metrics and energy consumption in a single‑machine, non‑preemptive environment. The authors consider two classic scheduling objectives—weighted completion time and weighted tardiness—augmented with a cost term that captures the energy (or more generally, any speed‑dependent) expense of executing each job. Unlike most prior work, which assumes a uniform, often polynomial, power‑speed relationship that is independent of the job, this study allows arbitrary non‑negative, job‑specific cost functions. This flexibility enables the framework to model not only CPU power but also maintenance, wear‑and‑tear, or replacement costs that may vary with both speed and job type.

The core technical contribution is a novel rounding scheme built on the well‑known α‑point technique, extended with a new concept called “α‑speeds.” The authors first formulate a time‑indexed integer program (IP) that assigns each job to a completion interval and a speed. By partitioning the time horizon into geometrically increasing intervals, they obtain a polynomial‑size linear programming (LP) relaxation. Solving this LP yields fractional assignments x_{i j t} indicating the fraction of job i processed at speed σ_j and completed in interval t.

From the LP solution, an α‑point for each job is defined as the earliest interval where an α‑fraction (typically α = 0.5) of the job’s processing has been completed, regardless of the speeds used. This α‑point determines a tentative order of jobs: sorting jobs by increasing α‑points (respecting precedence constraints) yields a feasible non‑preemptive sequence.

The second innovation, α‑speeds, interprets the distribution of a job’s processing across speeds as a probability mass function (pmf). The expected speed under this pmf is taken as the α‑speed at which the job will actually run. By collapsing the time dimension and using the LP’s fractional speed allocations, the authors guarantee that the total energy cost incurred by running each job at its α‑speed is at most a constant factor larger than the LP lower bound.

Combining the α‑point ordering with the α‑speed assignment, the authors design constant‑factor approximation algorithms for four problem variants:

1. Weighted completion time with release dates and precedence constraints.
2. Weighted completion time with only precedence constraints.
3. Weighted tardiness with deadlines and precedence constraints.
4. Weighted tardiness with only deadlines (no release dates).

For the weighted‑completion‑time case, the approximation ratios are 4(1+ε)(1+δ) when only precedence constraints are present, and (3+2√2)(1+ε)(1+δ) when release dates are also allowed. Here ε is the granularity of the interval discretization and δ bounds the ratio between consecutive speeds (σ_{j+1} ≤ (1+δ)σ_j).

For weighted tardiness, an additional regularity condition on the energy cost function is required: the cost must grow as s^{β−1} for some β ≥ 2. Under this assumption the algorithm achieves a ratio of 4β(1+ε)^{β−1}(1+δ)^{β−1}, even with arbitrary precedence constraints.

The paper further shows how to remove the assumption of a discrete speed set, extending the analysis to continuous speed ranges without loss of approximation quality. Moreover, the LP‑based framework naturally accommodates any non‑negative, job‑specific cost function, as long as it satisfies mild growth conditions, thereby broadening the applicability to scenarios such as equipment wear, maintenance, or time‑varying electricity pricing.

Although no empirical evaluation is presented, the theoretical results are significant: they provide the first constant‑factor approximations for energy‑aware weighted tardiness, a problem previously untouched, and improve upon earlier work that either required multiple machines, restricted speed sets, or could only handle linear combinations of energy and completion time.

In summary, the authors develop a powerful and versatile algorithmic toolkit that simultaneously optimizes traditional scheduling objectives and heterogeneous energy‑related costs. By leveraging α‑points for ordering and introducing α‑speeds for energy‑aware speed selection, they achieve provable, constant‑factor guarantees in a realistic setting that includes release dates, deadlines, and precedence constraints, and that allows highly general cost functions. This work advances both the theory of approximation algorithms for scheduling and the practical design of energy‑efficient computing systems.