Largest-Ratio-First Scheduling Algorithm

• The paper shows that the algorithm orders tasks by non-increasing Smith ratio and assigns each task to the worker with the least expected workload, optimizing total weighted completion time.
• It combines single-machine Smith ordering with parallel load balancing by incorporating expected communication delays between requesters and crowd workers.
• The study provides corrected approximation guarantees and integrates LRF into online frameworks like CosMOS, highlighting its strengths and limitations in identical-worker settings.

Searching arXiv for the specified scheduling papers to ground the article and cite them accurately. {"query":"arXiv (Chen, 6 Aug 2025) Approximation Algorithms for Scheduling Crowdsourcing Tasks in Mobile Social Networks"} {"query":"arXiv (Chen, 27 Mar 2026) Improved Algorithms for Unrelated Crowd Worker Scheduling in Mobile Social Networks"} {"query":"arXiv (Agnetis et al., 2019) The Largest-Z-ratio-First algorithm is 0.8531-approximate for scheduling unreliable jobs on m parallel machines"} Largest-Ratio-First (LRF) is a greedy list-scheduling algorithm for minimizing total weighted completion time on parallel workers. In the mobile social network formulation, a requester owns a set of indivisible crowdsourcing tasks, each task has a weight and a required service time, and each crowd worker incurs two expected requester-worker contact intervals, one for assignment and one for feedback. LRF orders tasks by non-increasing Smith ratio and assigns them one by one to the worker with the smallest current expected workload, thereby combining single-machine Smith ordering with parallel-machine load balancing under communication delay (Chen, 6 Aug 2025, Chen, 27 Mar 2026).

1. Formal scheduling model

In the mobile social network setting, the user set is $\mathcal{U}=\{u_0,u_1,\dots,u_m\}$, where $u_0$ is the requester and $u_j$ for $1\le j\le m$ are crowd workers. The requester has $n$ indivisible tasks $\mathcal{S}=\{s_1,s_2,\dots,s_n\}$. Each task has a required service time and a weight: in one notation these are $(\tau_i,w_i)$, and in another they are $(p_j,w_j)$ (Chen, 6 Aug 2025, Chen, 27 Mar 2026).

Communication is modeled through expected inter-contact times. In the rate-based notation, the inter-meeting time between requester $u_0$ and worker $u_j$ is exponential with rate $u_0$0, so the expected meeting time is $u_0$1. In the alternative notation, the expected inter-contact time for worker $u_0$2 is $u_0$3, and the total contact time for a task on that worker is $u_0$4 because each task requires one meeting for distribution and one meeting for feedback (Chen, 6 Aug 2025, Chen, 27 Mar 2026).

A schedule is a partition of tasks into worker-specific sets. Each worker processes its assigned tasks sequentially and non-preemptively. For a task assigned to worker $u_0$5, its completion time includes three components: one expected contact interval for assignment, the cumulative processing time of earlier tasks on that worker plus its own service time, and one expected contact interval for returning the result. The optimization objective is the exact total weighted completion time

$u_0$6

This is the quantity denoted by $u_0$7, with $u_0$8 for the optimum and $u_0$9 for the LRF schedule (Chen, 6 Aug 2025, Chen, 27 Mar 2026).

A central load quantity is the expected workload. For worker $u_j$0 with assigned task set $u_j$1,

$u_j$2

and if the worker has no assigned task, $u_j$3. In the $u_j$4 notation,

$u_j$5

with $u_j$6 if $u_j$7 (Chen, 6 Aug 2025, Chen, 27 Mar 2026).

2. Algorithmic definition

LRF uses the Smith ratio. In the identical-worker environment, where $u_j$8 for all workers, the priority index is

$u_j$9

equivalently $1\le j\le m$0 in the alternative notation. Tasks are presorted in non-increasing order of this ratio (Chen, 6 Aug 2025, Chen, 27 Mar 2026).

The offline algorithm initializes every worker with an empty task set and an expected workload equal to its two contact intervals. It then scans tasks in non-increasing Smith-ratio order. For each task, it selects a worker with minimum current expected workload,

$1\le j\le m$1

assigns the task to that worker, and updates the worker’s expected workload by adding the task’s service time. The resulting schedule is the partition of tasks across workers after all such assignments are made (Chen, 6 Aug 2025, Chen, 27 Mar 2026).

This construction has two structural consequences. First, the global ordering rule is purely ratio-based: high $1\le j\le m$2 tasks are considered earlier. Second, the assignment rule is purely load-based: each next task is sent to the currently least-loaded worker as measured by expected workload rather than by raw processing time. Within each worker, tasks are processed in assignment order, so the local sequence respects the global non-increasing Smith-ratio order. Tie-breaking for equal ratios or equal workloads is arbitrary (Chen, 6 Aug 2025).

For the identical-worker version presented in the unrelated-worker follow-up paper, the reported running time is $1\le j\le m$3: $1\le j\le m$4 for sorting and $1\le j\le m$5 for scanning workers to find the minimum expected workload at each assignment step (Chen, 27 Mar 2026).

3. Approximation guarantees and corrected analysis

LRF is closely connected to the classical identical-parallel-machine problem $1\le j\le m$6, for which Eastman et al. are recalled as establishing a $1\le j\le m$7-approximation for Largest Ratio First in the no-communication setting (Chen, 27 Mar 2026). In the mobile social network model, however, requester-worker contact times alter both lower-bound constructions and ratio analysis.

A central correction was made in "Approximation Algorithms for Scheduling Crowdsourcing Tasks in Mobile Social Networks" (Chen, 6 Aug 2025). That work shows that the approximation-ratio analysis presented by Zhang et al. for the communication-aware LRF algorithm is incorrect. Specifically, the lower bound

$1\le j\le m$8

is shown not to hold in general, via a counterexample with four equal-ratio tasks and two workers. The invalidity of that inequality means the previously claimed $1\le j\le m$9 approximation for offline LRF in the mobile social network model is not justified as stated (Chen, 6 Aug 2025).

The corrected general bound in that paper separates the processing component from the communication component. It derives

$n$0

The same work also proves a distinct bound in a service-time-dominant regime: when

$n$1

the approximation ratio can reach

$n$2

This argument uses a single-worker surrogate schedule and an averaging bound on task completion times (Chen, 6 Aug 2025).

A subsequent paper strengthens the communication-aware identical-worker guarantee. "Improved Algorithms for Unrelated Crowd Worker Scheduling in Mobile Social Networks" removes task-weight dependence from the ratio and proves

$n$3

where $n$4 and $n$5. The proof introduces communication-only terms

$n$6

and combines them with Eastman-type processing bounds through the inequality

$n$7

for positive $n$8 (Chen, 27 Mar 2026).

Setting	Approximation guarantee	Source
Communication-aware MSN, corrected general bound	$n$9	(Chen, 6 Aug 2025)
Service-time-dominant regime, $\mathcal{S}=\{s_1,s_2,\dots,s_n\}$0 and $\mathcal{S}=\{s_1,s_2,\dots,s_n\}$1	$\mathcal{S}=\{s_1,s_2,\dots,s_n\}$2	(Chen, 6 Aug 2025)
Improved identical-worker communication-aware bound	$\mathcal{S}=\{s_1,s_2,\dots,s_n\}$3	(Chen, 27 Mar 2026)

These results locate the main theoretical sensitivity of LRF in the communication model. The later improvement indicates that, in the identical-worker environment, communication heterogeneity rather than weight heterogeneity is the relevant worst-case parameter (Chen, 27 Mar 2026).

4. Online use through CosMOS

LRF also appears as the offline engine inside the online framework CosMOS. In that framework, whenever the requester meets a worker at time $\mathcal{S}=\{s_1,s_2,\dots,s_n\}$4, the algorithm initializes a reduced offline instance on the remaining unscheduled tasks and remaining workers, sets the currently met worker’s workload to its total contact time $\mathcal{S}=\{s_1,s_2,\dots,s_n\}$5, sets every other remaining worker’s workload to $\mathcal{S}=\{s_1,s_2,\dots,s_n\}$6, runs an offline algorithm such as LRF, assigns to the met worker the tasks allocated to it in that hypothetical offline solution, schedules those tasks locally in Smith-ratio order, and then removes both the assigned tasks and that worker from the residual instance (Chen, 27 Mar 2026).

The corrected 2025 paper gives an online competitive-ratio template of the form

$\mathcal{S}=\{s_1,s_2,\dots,s_n\}$7

where $\mathcal{S}=\{s_1,s_2,\dots,s_n\}$8 is taken from the offline LRF analysis: in general,

$\mathcal{S}=\{s_1,s_2,\dots,s_n\}$9

and under $(\tau_i,w_i)$0,

$(\tau_i,w_i)$1

This expresses the online loss as the offline approximation multiplied by a factor depending on communication relative to total service time (Chen, 6 Aug 2025).

The later 2026 paper provides an updated competitive ratio for CosMOS under an offline algorithm with approximation factor $(\tau_i,w_i)$2:

$(\tau_i,w_i)$3

Substituting the improved LRF bound gives an explicit online guarantee for LRF-based CosMOS in the identical-worker environment (Chen, 27 Mar 2026).

5. Extensions to unrelated workers and empirical behavior

LRF itself is defined for the identical-worker environment. In the unrelated-worker model, the service time of task $(\tau_i,w_i)$4 depends on the assigned worker, so $(\tau_i,w_i)$5 is arbitrary over $(\tau_i,w_i)$6 rather than equal to a worker-independent $(\tau_i,w_i)$7 (Chen, 27 Mar 2026). The main algorithms proposed for that setting are not direct LRF extensions; instead they are time-indexed LP-based methods: RTS, DTS, and EDTS (Chen, 27 Mar 2026).

For empirical comparison in the unrelated setting, three LRF-like heuristics are defined by replacing each task’s worker-dependent service times with a proxy processing time:

• LRF-MAX: $(\tau_i,w_i)$8.
• LRF-MIN: $(\tau_i,w_i)$9.
• LRF-MEAN: $(p_j,w_j)$0.

Each heuristic sorts tasks by non-increasing proxy Smith ratio $(p_j,w_j)$1 and then assigns each task to the worker that minimizes its completion time under the current partial assignment (Chen, 27 Mar 2026).

Performance is evaluated by the weighted completion time ratio

$(p_j,w_j)$2

so smaller values are better and values close to $(p_j,w_j)$3 are near the LP lower bound. Across synthetic datasets and real MSN traces from Cambridge Haggle, Intel, and Infocom, EDTS is reported to consistently outperform the three LRF variants. The gap widens as task density increases, and EDTS also shows lower variance. At the same time, the experiments report that as the influence-factor ratio approaches $(p_j,w_j)$4, making the instance nearly uniform and closer to a related-machine environment, the WCTR of LRF variants drops and they can outperform EDTS (Chen, 27 Mar 2026).

These results support a sharp distinction. In identical-worker settings with moderate communication heterogeneity, LRF has a concise ratio rule and provable guarantees. In genuinely heterogeneous unrelated-worker settings, LP-based assignment plus per-worker Smith ordering yields better theoretical and empirical performance (Chen, 27 Mar 2026).

6. Relation to Smith’s rule and to other ratio-based algorithms

The organizing principle behind LRF is Smith’s rule. For a fixed machine or worker, once the assigned task set is known, ordering tasks in non-increasing $(p_j,w_j)$5 is optimal for minimizing $(p_j,w_j)$6. Both the LRF algorithm and the unrelated-worker algorithms DTS and EDTS exploit this fact: the principal difficulty lies in assignment across workers, whereas the optimal local sequencing rule is already determined by Smith ratios (Chen, 27 Mar 2026).

This places LRF within a broader family of ratio-based scheduling algorithms, but not all similarly named procedures optimize the same objective. A notable example is the Largest-Z-ratio-First algorithm for scheduling unreliable jobs on $(p_j,w_j)$7 parallel machines. There, each job has success probability $(p_j,w_j)$8 and reward $(p_j,w_j)$9, the objective is to maximize expected reward rather than minimize weighted completion time, and the priority index is

$u_0$0

Jobs are sorted by non-increasing $u_0$1 and assigned to the machine with maximum cumulative probability, not the worker with minimum expected workload. That algorithm is proved to be a $u_0$2-approximation, and the bound is tight (Agnetis et al., 2019).

The distinction is substantive. LRF is a completion-time minimization rule built around the Smith ratio and additive workloads; Largest-Z-ratio-First is an expected-reward maximization rule built around a failure-risk ratio and multiplicative survival probabilities. The similarity in naming reflects a common ratio-ordering paradigm, but the models, objectives, and analyses are different (Agnetis et al., 2019).

7. Significance and limitations

LRF remains important because it is algorithmically simple, structurally transparent, and analytically tractable. In its canonical form, it needs only a Smith-ratio ordering and a running estimate of each worker’s expected workload. This makes it a natural baseline for mobile crowdsourcing systems in which assignment and feedback are both gated by requester-worker meetings (Chen, 6 Aug 2025, Chen, 27 Mar 2026).

Its limitations are equally clear in the current literature. First, the classical $u_0$3-style reasoning does not transfer naively once communication terms are inserted into lower bounds; this is the central correction made in 2025 (Chen, 6 Aug 2025). Second, even after that correction, worst-case behavior depends on communication heterogeneity, and in the earlier corrected analysis it also depended on task-weight heterogeneity (Chen, 6 Aug 2025). Third, in unrelated-worker environments there is no formally analyzed direct LRF extension in the cited work; the available LRF-MAX, LRF-MIN, and LRF-MEAN procedures are heuristics used as baselines, not proven approximations (Chen, 27 Mar 2026).

The current state of the topic therefore presents LRF as a rigorously studied greedy algorithm for identical-worker, communication-aware weighted-completion scheduling, with corrected and improved approximation guarantees, a clear online integration through CosMOS, and a well-defined boundary beyond which LP-based methods become preferable (Chen, 6 Aug 2025, Chen, 27 Mar 2026).