Column matching is a central task in reconciling schemas for data integration. Column names and descriptions are valuable for this task. LLMs can leverage such natural-language schema metadata. However, in many datasets, correct matching requires additional evidence beyond the column itself. Because it is impractical to provide an LLM with the entire schema metadata needed to capture this evidence, the core challenge becomes to select and organize the most useful contextual information. We present ConStruM, a structure-guided framework for budgeted evidence packing in schema matching. ConStruM constructs a lightweight, reusable structure in which, at query time, it assembles a small context pack emphasizing the most discriminative evidence. ConStruM is designed as an add-on: given a shortlist of candidate targets produced by an upstream matcher, it augments the matcher's final LLM prompt with structured, query-specific evidence so that the final selection is better grounded. For this purpose, we develop a context tree for budgeted multi-level context retrieval and a global similarity hypergraph that surfaces groups of highly similar columns (on both the source and target sides), summarized via group-aware differentiation cues computed online or precomputed offline. Experiments on real datasets show that ConStruM improves matching by providing and organizing the right contextual evidence.

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