CaWoSched: Carbon-Aware Workflow Scheduling

• CaWoSched is a carbon-aware workflow scheduling framework that optimizes task timing on pre-assigned workflows to reduce carbon emissions under strict deadlines.
• It employs greedy heuristics combined with local search, using slack and pressure score functions to select optimal time intervals with high green power availability.
• Empirical results show significant carbon savings over ASAP baselines, establishing CaWoSched as a robust benchmark for subsequent carbon-aware scheduling research.

Searching arXiv for the original CaWoSched paper and its follow-up comparator paper. CaWoSched is a carbon-aware workflow scheduling framework for deadline-constrained workflows with fixed mapping and fixed ordering. It addresses the temporal scheduling problem that remains after tasks and communications have already been assigned to processors and ordered—“say using the well-known HEFT algorithm”—by shifting execution across time intervals with different green power budgets so as to minimize carbon emissions while maintaining the deadline (Schweisgut et al., 11 Jul 2025). Within the literature summarized here, it is presented both as the original heuristic framework for this restricted problem and as the state-of-the-art comparator against which later joint mapping-and-scheduling methods are evaluated (Schweisgut et al., 26 May 2026).

1. Definition and problem scope

CaWoSched is formulated for workflows represented as directed acyclic graphs under a mixed renewable and non-renewable energy supply. The central assumption is restrictive but explicit: “the mapping and ordering of the workflow tasks are given ... together with a time deadline to complete the execution,” and the remaining decision problem is to “shift tasks across carbon intervals on each processor so as to minimize carbon emissions while maintaining the deadline” (Schweisgut et al., 26 May 2026). In this sense, CaWoSched is not a placement framework. It does not decide where tasks run, and it does not change the local task order on a processor. Its optimization variable is the timing of already mapped and ordered computations and communications (Schweisgut et al., 11 Jul 2025).

The workflow model follows a weighted DAG $G=(V,E,\omega,c)$, where $V$ is the set of tasks, $E$ is the set of precedence constraints, $\omega(v)$ is task work, and $c(v_i,v_j)$ is communication volume or communication time depending on the formulation (Schweisgut et al., 26 May 2026, Schweisgut et al., 11 Jul 2025). In the broader notation used by later work that explicitly follows CaWoSched, heterogeneous processors have speeds $s(p_k)$, so a task $v_i$ executes in time

$t(v_i)=\frac{\omega(v_i)}{s(\mu(v_i))}.$

Communications are modeled similarly to CaWoSched as extra tasks on communication channels (Schweisgut et al., 26 May 2026).

The time horizon is partitioned into intervals with a constant green power budget per interval. Carbon cost is incurred only when the total platform power exceeds the green budget of the current interval. Thus CaWoSched targets carbon minimization under a makespan bound rather than energy minimization or a mixed energy-time objective (Schweisgut et al., 26 May 2026).

2. System model and optimization objective

CaWoSched operates in a scheduling model where workflow precedence, processor order, communications, and deadline feasibility are explicit. In the shared formalization later reused by the follow-up work, the objective is “to find a valid mapping and schedule in order to minimize the carbon cost, while not exceeding a bound $D$ on the makespan,” with makespan constraint

$\max_{v\in V}\{\sigma(v)+ t(v)\}\leq D.$

For CaWoSched specifically, the mapping is fixed, so only the scheduling component remains variable (Schweisgut et al., 26 May 2026).

Communications are represented as extra tasks. For an edge $V$0, there is a communication task $V$1. If $V$2, communication is mapped to the corresponding link and takes time

$V$3

with $V$4 in the later paper’s experiments; otherwise, communication cost is zero (Schweisgut et al., 26 May 2026). This communication-as-task representation is also explicit in the original CaWoSched paper through the communication-enhanced DAG $V$5, where communication tasks are added on fictional communication processors (Schweisgut et al., 11 Jul 2025).

Schedule validity includes nonnegativity, processor exclusivity, and precedence. In the later formulation aligned with CaWoSched, these conditions are: $V$6 and for tasks on the same processor,

$V$7

while precedence is enforced either directly for same-processor tasks or via inserted communication tasks for inter-processor edges (Schweisgut et al., 26 May 2026). In the original CaWoSched formulation, processor non-overlap is absorbed into the communication-enhanced DAG by adding order edges on each processor and communication processor (Schweisgut et al., 11 Jul 2025).

Green-power availability is modeled over a horizon $V$8 partitioned into intervals $V$9, each with green budget $E$0 (Schweisgut et al., 26 May 2026). In the original paper, total carbon cost is

$E$1

where for time $E$2,

$E$3

The objective is therefore the minimization of excess power over available green supply (Schweisgut et al., 11 Jul 2025).

3. Algorithmic structure and heuristic variants

CaWoSched is a heuristic framework “that combines several greedy approaches with local search” (Schweisgut et al., 11 Jul 2025). The core algorithmic picture is consistent across the original paper and its later use as a baseline: it takes a given mapping and order, shifts tasks and communications toward intervals with high green power budget, and uses local search for refinement (Schweisgut et al., 26 May 2026).

The original framework computes timing windows using earliest and latest start times. For source tasks,

$E$4

and otherwise

$E$5

For sink tasks,

$E$6

and otherwise

$E$7

These induce slack

$E$8

and pressure

$E$9

Slack-based variants favor tasks with little timing flexibility; pressure-based variants favor tasks with large runtime relative to flexibility (Schweisgut et al., 11 Jul 2025).

The framework combines four score families—slack, slackW, press, pressW—with two interval granularities—normal and refined—to obtain eight greedy variants:

• slack
• slackW
• slackR
• slackWR
• press
• pressW
• pressR
• pressWR

Each of these can be augmented with local search, yielding sixteen heuristics in total (Schweisgut et al., 11 Jul 2025). The refined interval option uses a finer subdivision derived from aligning blocks of up to $\omega(v)$0 consecutive tasks to original interval starts and ends (Schweisgut et al., 11 Jul 2025).

Greedy placement considers interval starts $\omega(v)$1 within the legal window

$\omega(v)$2

If such an interval exists, the heuristic chooses a feasible interval with the highest remaining green budget, breaking ties by earliest start (Schweisgut et al., 11 Jul 2025). After placement, green budgets are updated and timing windows for unscheduled tasks are recomputed or updated (Schweisgut et al., 11 Jul 2025).

Local search is processor-oriented and move-based. Processors are sorted by nonincreasing power consumption, and tasks are scanned left to right. For each task, the method checks moves up to $\omega(v)$3 time units to the left and right, with $\omega(v)$4 in the experiments, accepting the first improving legal move. Because it is a hill-climbing improvement phase, it never worsens the schedule (Schweisgut et al., 11 Jul 2025).

The later paper identifies two CaWoSched variants as experimental baselines:

• H-CWS-p
• H-CWS-s

It states that “-p and -s represent the base score pressure and slack, respectively,” and that “$\omega(v)$5 and $\omega(v)$6 perform similarly -- with $\omega(v)$7 slightly better” (Schweisgut et al., 26 May 2026). The later text does not further unpack the internal meaning of “pressure” and “slack” beyond their names, but this is consistent with the original pressure- and slack-based score functions (Schweisgut et al., 11 Jul 2025).

4. Complexity and theoretical position

The theoretical contribution of the original CaWoSched paper is a complexity split between the uniprocessor and multiprocessor cases. With a single processor and fixed task order, the problem is polynomial-time solvable (Schweisgut et al., 11 Jul 2025). The proof proceeds via a pseudo-polynomial dynamic program over task completion times and then shows that there always exists an optimal $\omega(v)$8-schedule aligned with interval boundaries, which yields a polynomial-time algorithm (Schweisgut et al., 11 Jul 2025).

For the uniprocessor case, if tasks $\omega(v)$9 execute in fixed order, the dynamic program considers

$c(v_i,v_j)$0

as the minimum cost for the first $c(v_i,v_j)$1 tasks when task $c(v_i,v_j)$2 completes exactly at time $c(v_i,v_j)$3, with recurrence

$c(v_i,v_j)$4

The structural result that an optimal boundary-aligned schedule exists is then used to restrict the set of candidate completion times (Schweisgut et al., 11 Jul 2025).

For at least two processors, the problem becomes strongly NP-hard even with uniform processors and independent tasks (Schweisgut et al., 11 Jul 2025). The reduction is from 3-Partition and uses alternating green and zero-green intervals to force a zero-carbon schedule to correspond to an exact partition (Schweisgut et al., 11 Jul 2025).

Later work sharpens the contrast between CaWoSched’s restricted setting and joint mapping-and-scheduling formulations. It states that in the original CaWoSched work, “the decision problem with independent tasks (hence, no communication) that are already mapped and ordered on a set of homogeneous processors ... is strongly NP-complete” (Schweisgut et al., 26 May 2026). It also notes that the more general problem with mapping and ordering decisions becomes strongly NP-complete even on a single processor and admits no constant-factor approximation unless $c(v_i,v_j)$5, whereas the single-processor fixed-mapping-and-ordering case is polynomial in CaWoSched (Schweisgut et al., 26 May 2026). This establishes CaWoSched as a deliberately restricted but algorithmically more tractable formulation.

5. Empirical performance and comparative evaluation

The original paper evaluates CaWoSched on 34 workflows, two heterogeneous clusters, and 16 green-power profiles per workflow-cluster pair, for $c(v_i,v_j)$6 instances per algorithm (Schweisgut et al., 11 Jul 2025). Workflows range from 200 to 30,000 tasks and include both real and synthetic instances derived from atacseq, bacass, eager, and methylseq (Schweisgut et al., 11 Jul 2025). Mappings are generated by HEFT, deadlines are set to $c(v_i,v_j)$7, $c(v_i,v_j)$8, $c(v_i,v_j)$9, and $s(p_k)$0 where $s(p_k)$1 is the makespan of an ASAP baseline, and the green-power scenarios include concave, convex, sinusoidal, and roughly constant profiles (Schweisgut et al., 11 Jul 2025).

The baseline in the original paper is ASAP, which schedules every task at its earliest feasible start time and ignores green-energy variation (Schweisgut et al., 11 Jul 2025). Against this baseline, all CaWoSched variants provide substantial carbon savings. ASAP is worst in $s(p_k)$2 of instances, and the best median heuristic-to-baseline ratio is $s(p_k)$3 for pressWR-LS (Schweisgut et al., 11 Jul 2025). The paper also highlights instances where gains are much larger under looser deadlines; for example, slackW reaches a cost ratio of $s(p_k)$4 in one case (Schweisgut et al., 11 Jul 2025).

No single heuristic dominates universally, but the paper reports that pressWR-LS is ranked first in $s(p_k)$5 of instances (Schweisgut et al., 11 Jul 2025). Pressure-based variants perform especially well under tight deadlines, while slack-based variants tend to benefit more when deadlines are looser (Schweisgut et al., 11 Jul 2025). Local search is particularly important: the average cost ratio between LS-enhanced and non-LS solutions is around $s(p_k)$6–$s(p_k)$7, and many instances are improved to zero carbon cost by local search although the greedy schedule still has positive cost (Schweisgut et al., 11 Jul 2025).

The original paper also provides an exact pseudo-polynomial time-indexed ILP for small instances. It minimizes

$s(p_k)$8

with binary variables for task start, end, and running status over time, but becomes too slow beyond about 200 tasks (Schweisgut et al., 11 Jul 2025). On small instances, CaWoSched heuristics often match the optimum or come close to it, while running in milliseconds or seconds instead of up to one hour for the ILP (Schweisgut et al., 11 Jul 2025).

The later paper on joint carbon-aware mapping and scheduling uses CaWoSched as the main state-of-the-art comparator (Schweisgut et al., 26 May 2026). In that evaluation, the compared algorithms are:

• CWM
• H-CWS-p
• H-CWS-s
• HEFT-SL

The workloads comprise 44 workflows from atacseq, bacass, methylseq, eager, and chipseq, scaled to 12–30000 vertices, on clusters with 72 and 144 nodes and on Germany 2024 and California 2024 Electricity Maps hourly carbon data transformed into green-power budgets (Schweisgut et al., 26 May 2026). The total number of instances is

$s(p_k)$9

At deadline $v_i$0, where $v_i$1 is the makespan of the carbon-agnostic HEFT-SL baseline, the abstract reports that the new method achieves a median carbon cost reduction of $v_i$2 over the best version of CaWoSched, and that “CaWoSched itself already reduces the carbon-agnostic baseline by $v_i$3” (Schweisgut et al., 26 May 2026). More specifically, at $v_i$4, CWM reduces median carbon cost by $v_i$5 versus $v_i$6 and $v_i$7 versus $v_i$8 (Schweisgut et al., 26 May 2026). At $v_i$9, however, the paper states that the median improvement over CaWoSched is rather small, indicating that CaWoSched remains competitive when deadline slack is tight (Schweisgut et al., 26 May 2026).

6. Interpretation, limitations, and place in the literature

CaWoSched’s principal strength is that it is already a strong carbon-aware scheduler under realistic workflow constraints. The later paper explicitly calls it “the only competitor we know of that also optimizes for carbon cost within a comparable framework, even though it further considers that the mapping and ordering is fixed” (Schweisgut et al., 26 May 2026). It handles workflows with precedence constraints and explicit communications, and it is computationally lighter than later joint mapping-and-scheduling methods because it “only shift[s] tasks within a given mapping” (Schweisgut et al., 26 May 2026).

Its main limitation is also explicit: mapping and ordering are fixed (Schweisgut et al., 26 May 2026, Schweisgut et al., 11 Jul 2025). This restriction is central to both its practicality and its performance ceiling. In heterogeneous clusters with time-varying renewable supply, later work argues that “it is equally important where the tasks are executed,” and this is exactly the dimension that CaWoSched does not optimize (Schweisgut et al., 26 May 2026). A plausible implication is that CaWoSched is best understood as a temporal carbon-optimization layer that can refine the output of a prior mapper such as HEFT, rather than as a full workflow placement-and-scheduling system.

The framework is also sensitive to the amount of slack. Under tight deadlines, CaWoSched and later joint methods are close to one another, and for tiny workflows most algorithms often find a carbon-optimal schedule (Schweisgut et al., 26 May 2026). Under looser deadlines, larger workflows, and larger clusters, the advantage of joint mapping and scheduling becomes clearer (Schweisgut et al., 26 May 2026). This suggests that CaWoSched’s fixed-mapping design limits its ability to exploit additional flexibility in heterogeneous environments.

The original bibliographic reference given by the later paper identifies the prior work as: Dominik Schweisgut, Anne Benoit, Yves Robert, and Henning Meyerhenke.

Carbon-Aware Workflow Scheduling with Fixed Mapping and Deadline Constraint. In Proc. of the 54th Int. Conf. on Par. Proc. (ICPP 2025), pages 627--637, ACM, 2025 (Schweisgut et al., 26 May 2026). The corresponding arXiv preprint is “Carbon-Aware Workflow Scheduling with Fixed Mapping and Deadline Constraint” (Schweisgut et al., 11 Jul 2025).

Taken together, these sources position CaWoSched as a specialized workflow scheduler for minimizing carbon cost by exploiting time-varying green-power availability after mapping and ordering decisions have already been made. Its theoretical profile is unusually sharp for a workflow carbon scheduler: polynomial on one processor, strongly NP-hard on multiple processors (Schweisgut et al., 11 Jul 2025). Its practical profile is that of a robust heuristic framework whose greedy-plus-local-search variants consistently outperform a carbon-agnostic ASAP baseline and provide a meaningful state-of-the-art baseline for subsequent work on more general carbon-aware workflow optimization (Schweisgut et al., 11 Jul 2025, Schweisgut et al., 26 May 2026).