Relevance Is Not Necessity: Selecting the Necessary API Set for Tool-Using LLMs

A multi-tool plan is correct only when its call set matches the gold API set exactly. Standard tool-use pipelines optimize first-hop relevance, but downstream verifiers require the necessary API set, creating a relevance--necessity gap. This study formalizes the gap: under correlated API indicators, symmetric decomposable selectors based only on per-API marginals can be dominated by a Bayes-optimal set selector, while whole-set exact-match supervision is a proper scoring rule for the optimal selector restricted to a candidate menu. We propose an offline refinement framework for frozen upstream traces. Under a leakage-free ToolBench G2/G3 protocol, TC-MASS improves verifier-consistent API-set recovery without additional LLM generation after the upstream trace exists. Ablations and cross-distribution probes support the central claim: necessity-aware set-level supervision, not relevance ranking alone, is the right objective for fixed-pool API-set recovery.