用张量分解让多个智能体的计划自动避冲突，无需人工规则。
Tensor-Coord: Algebraic Decomposition of Joint Plan Tensors for Conflict-Free Multi-Agent LLM Planning
- 把多智能体计划建模为三维张量，用代数方法挖掘协作结构。
- 2-4个智能体在5×5网格上，平均1.4-4.0轮内实现无冲突规划。
- 可解释角色与阶段，自动生成自然语言约束供模型迭代优化。
大语言模型在多智能体规划中受限于独立生成计划引发的协调失败，如空间碰撞、资源争用和时间死锁。本文提出Tensor-Coord，一种基于多重线性代数的框架，将N个智能体的联合计划表示为三阶张量 $T \in R^{N \times H \times A}$，涵盖智能体、时间步和动作。通过标准型（CP）与托克尔（Tucker）分解识别潜在协调结构。最小ε近似CP秩 $R^*$ 定义了可计算的协调复杂度度量，$CC(Pi)=(R^*-N)/N$。证明 $R^*=N$ 是计划独立性的充要条件。残差 $E=T-T_{R^*}$ 构成冲突得分，定位失败而无需领域规则。托克尔因子提供可解释的智能体角色、时间阶段与动作聚类，并转化为自然语言约束用于迭代重规划。在多机器人配送任务中，2个智能体（5×5网格）100%收敛，平均1.4轮；3个智能体80%成功，平均3.2轮；4个智能体60%成功，平均4.0轮。CP秩近似线性增长：$R*(N) = 3.9N + 0.5$，支持其作为协调复杂度预测器。
原文摘要 · Abstract (English)
Large language models (LLMs) remain limited in multi-agent planning because independently generated plans can create coordination failures such as spatial collisions, resource contention, and temporal deadlocks. We introduce Tensor-Coord, a multilinear algebra framework that represents the joint plan of N agents as a third-order tensor \(T \in R^{N \times H \times A}\) over agents, timesteps, and actions. Canonical Polyadic (CP) and Tucker decompositions are used to identify latent coordination structure. The minimal epsilon-approximate CP rank R* defines a computable coordination complexity measure, with \(CC(Pi)=(R*-N)/N\). We prove that R*=N is necessary and sufficient for plan independence. The residual \(E=T-T_{R*}\) defines a conflict score over agent pairs, timesteps, and actions, localizing failures without domain-specific rules. Tucker factors provide interpretable agent roles, temporal phases, and action clusters that are converted into natural language constraints for iterative LLM replanning. Experiments on multi-robot delivery tasks across Easy (2 agents, 5x5 grid), Medium (3 agents, 5x5 grid), and Hard (4 agents, 5x5 grid) settings show convergence to conflict-free plans in 100% of 2-agent cases within 1.4 iterations on average, 80% of 3-agent cases within 3.2 iterations, and 60% of 4-agent cases within 4.0 iterations. CP rank scaled approximately linearly as \(R*(N) = 3.9N + 0.5\), supporting its use as a predictor of coordination complexity.
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