Tensor-Coord: Algebraic Decomposition of Joint Plan Tensors for Conflict-Free Multi-Agent LLM Planning

• URL: http://arxiv.org/abs/2606.16478v1
• Date: Mon, 15 Jun 2026 09:44:26 GMT
• Title: Tensor-Coord: Algebraic Decomposition of Joint Plan Tensors for Conflict-Free Multi-Agent LLM Planning
• Abstract summary: Large language models (LLMs) remain limited in multi-agent planning because independently generated plans can create coordination failures.<n>We introduce 3.9-Coord, a multilinear algebra framework that represents the joint plan of N agents as a third-order tensor (T in RN times H times A) over agents, timesteps, and actions.
• Score: 0.0
• License: http://creativecommons.org/publicdomain/zero/1.0/
• Abstract: 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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