Structured Personalization: Modeling Constraints as Matroids for Data-Minimal LLM Agents

Personalizing Large Language Model (LLM) agents requires conditioning them on user-specific data, creating a critical trade-off between task utility and data disclosure. While the utility of adding user data often exhibits diminishing returns (i.e., submodularity), enabling near-optimal greedy selection, real-world personalization is complicated by structural constraints. These include logical dependencies (e.g., selecting fact A requires fact B), categorical quotas (e.g., select at most one writing style), and hierarchical rules (e.g., select at most two social media preferences, of which at most one can be for a professional network). These constraints violate the assumptions of standard subset selection algorithms. We propose a principled method to formally model such constraints. We introduce a compilation process that transforms a user's knowledge graph with dependencies into a set of abstract macro-facets. Our central result is a proof that common hierarchical and quota-based constraints over these macro-facets form a valid laminar matroid. This theoretical characterization lets us cast structured personalization as submodular maximization under a matroid constraint, enabling greedy with constant-factor guarantees (and (1-1/e) via continuous greedy) for a much richer and more realistic class of problems.