Chain-of-Scheduling (CoS) Framework Overview

• Chain-of-Scheduling (CoS) is a dual interpretation framework that addresses both event-based social network scheduling and online task allocation with optimal utility and interpretability.
• It employs a three-stage decomposition—exploration, verification, and integration—to efficiently generate, validate, and select high-utility candidate schedules.
• Empirical results demonstrate that CoS achieves near-optimal utility with reduced computation time and lower conflict rates compared to traditional and unstructured methods.

The Chain-of-Scheduling (CoS) framework encompasses both a structured reasoning protocol for event scheduling in Event-Based Social Networks (EBSNs), utilizing LLMs, and a body of algorithms for online scheduling of partially ordered tasks to processors. CoS frameworks aim at resolving event sequencing and processor assignment problems, focusing on optimality, interpretability, and computational efficiency, in contexts ranging from utility-maximizing user recommendations to on-line multiprocessor scheduling under precedence constraints (Zhao et al., 17 Nov 2025, Bosek, 2018).

1. Formal Problem Settings and Motivation

In the EBSN context, the CoS framework addresses the task of recommending event schedules that maximize user-specific utilities subject to temporal and geographic feasibility. Events are modeled as $$e_i = \langle \text{loc}_i, t_i^{\text{start}}, t_i^{\text{end}} \rangle$$, and each user $u_j$ associates a utility score $s_{i,j} \in [0,1]$ with every event. A feasible schedule $$T = \langle e_{i_1} \rightarrow ... \rightarrow e_{i_m} \rangle$$ must satisfy $$t_i^{\text{end}} + t(\text{loc}_i,\text{loc}_{i'}) \leq t_{i'}^{\text{start}}$$ for consecutive events, with the objective: $$T^* = \arg\max_{T \text{ valid}} \sum_{e \in T} s_{e,j}$$ The solution space, denoted $\mathbb{T}$, is exponentially large due to the NP-hardness of the underlying scheduling problem, formally established via a reduction from Directed Hamiltonian Path (Zhao et al., 17 Nov 2025).

In online processor allocation, a related interpretation arises: given a poset $(P, \leq)$ of "tasks" with arbitrary partial order, each task is assigned irrevocably and immediately to a processor (modeled as a "chain") when it arrives. The objective is to minimize the required number of processors while obeying all precedence constraints, with various structural results providing processor bounds based on the poset's width (Bosek, 2018).

2. The Three-Stage Decomposition in CoS

Central to the LLM-based CoS framework is its decomposition of schedule reasoning into three atomic, sequential stages:

1. Exploration: Efficiently enumerate a limited set ($k$) of high-utility, valid schedules. This is achieved by deploying combinatorial solvers (e.g., dynamic programming, grid search) to find the top-$k$ candidates $$T_{\text{top-}k} = \arg\text{top-}k_{T \in \mathbb{T}} \sum_{e_i \in T} s_{i,j}$$.
2. Verification: Explicitly compute the total utility for each candidate $T \in T_{\text{top-}k}$ as $v(T) = \sum_{e_i \in T} s_{i,j}$. The LLM is trained to verify and compare these aggregate scores.
3. Integration: Select the candidate $T^* = \arg\max_{T \in T_{\text{top-}k}} v(T)$ as the optimal schedule among those explored, ensuring global optimality (within the candidate set).

This decomposition forms a "CoS trace"—a deterministic, interpretable reasoning chain that avoids redundant steps and ensures traceable, high-quality solutions. Empirical ablation (removal of any stage) consistently reduces utility or increases scheduling conflicts, demonstrating the compositional necessity of all three stages (Zhao et al., 17 Nov 2025).

3. Distillation of CoS into LLMs

The CoS framework leverages knowledge distillation to internalize the three-stage protocol within LLMs, enabling autonomous, interpretable schedule reasoning:

• Teacher Models: Generate (prompt $\to$ CoS trace) pairs offline using exact or approximate combinatorial solvers.
• Student LLMs: Base models (e.g., Qwen2.5-7B-Instruct, Mistral-7B) are supervised-finetuned on these traces.

The construction of the supervised-finetuning dataset $$\mathcal{D}_{\text{SFT}} = \{(x_i, y_i^{\text{CoS}})\}_{i=1}^N$$ encodes both the event-user data (input $x_i$) and full CoS chain (output $y_i^{\text{CoS}}$). The objective function is cross-entropy over CoS tokens: $$\mathcal{L}_{\text{SFT}}^{(i)}(\theta) = - \sum_{t=1}^{L_i} \log P_{\text{LLM}}(y_{i,t}^{\text{CoS}} \mid x_i, y_{i,<t}^{\text{CoS}}; \theta)$$ Hyperparameters include low-rank adaptation (LoRA, with $\alpha=16$, rank $r=8$), context window of 32,768 tokens, and a learning rate of $10^{-5}$, trained for three epochs with small batch size. Two NVIDIA A800-SXM4-80 GB GPUs are used for training (Zhao et al., 17 Nov 2025).

Upon completion, the LLM generates full CoS traces in under 2 seconds per query, correctly enforcing all spatio-temporal and utility maximization constraints.

4. Theoretical Guarantees and Combinatorial Underpinnings

The underlying scheduling problems addressed by CoS are NP-hard, as demonstrated by a reduction from the Directed Hamiltonian Path (Appendix B in (Zhao et al., 17 Nov 2025)). No polynomial-time method can assure globally optimal results in the general case. Nonetheless, empirical evaluations indicate CoS achieves 90%–100% of the optimal utility on real-world datasets.

Classical on-line chain partitioning perspectives, as in (Bosek, 2018), frame scheduling as irreversible task-to-processor assignments in posets:

• General poset width $w$: Kierstead's algorithm guarantees coverage with $K(w) = (5^w-1)/4$ chains.
• Width-2 posets: 5 chains suffice.
• Width-3 posets: 16 chains improve the previous 31-chain bound.
• Up-growing interval orders: Tight result at $2w-1$ chains. Dilworth's Theorem ensures any finite poset can be partitioned into $w$ chains, but the on-line model incurs exponential overhead in the worst case for general $w$. Structured special cases permit substantial efficiency gains.

These combinatorial insights underpin the Exploration phase in LLM-based CoS, where efficient candidate generation is crucial for tractability and solution quality.

5. Empirical Evaluation and Interpretability

Performance benchmarks on real-world Meetup datasets (New York, Washington, London) substantiate CoS's claims of high utility, low latency, and improved validity:

Method	Utility (NY/Wash/Lon)	Latency (s)	Conflict Rate (%)
Exact Search (DP)	3.73/4.02/5.17	8–24	0
Greedy/GA	≤2.5	≪1	-
GOOSE, GNN-DRL	2.85–3.15	0.5–60	-
Off-the-shelf LLMs (DeepSeek)	≤2.7	30–785	86–99
CoS (Qwen2.5-7B)	3.37/3.57/4.50	1.29–3.36	15–41

CoS approaches the effectiveness of dynamic programming (DP) while requiring only 1–5 seconds computation and drastically reducing scheduling conflicts compared to unstructured LLM outputs. Interpretability is guaranteed: CoS traces enumerate candidates, enumerate utility computations, and show explicit reasoning at each decision step. Side-by-side comparisons with baselines highlight the transparency and auditable nature of the method (Zhao et al., 17 Nov 2025).

6. Zero-Shot Generalization and Synthesis with Online Partitioning

CoS demonstrates transferability of learned schedule reasoning. LLMs trained solely on New York scheduling data exhibit robust zero-shot performance on Washington and London, with utility gains of up to 50% over GOOSE and GNN-DRL baselines. This provides evidence that distilled CoS traces encode universalizable spatio-temporal scheduling semantics.

The synthesis with classical online chain partitioning is manifest in the shared paradigm: tasks/events with partially ordered constraints are assigned to chains/processors in a sequential—often irrevocable—manner. In both the modern LLM and classical settings, the CoS framework establishes the protocol for assigning tasks to resources, leveraging stagewise reasoning or processor-bound guarantees appropriate to the domain context (Zhao et al., 17 Nov 2025, Bosek, 2018).