Sparsest navigable graph construction is approximation-equivalent to Set Cover, yielding a (ln n+1)-approximation, o(ln n)-hardness, fast bicriteria algorithms, and an Ω(n squared) query lower bound.
Beyond the Worst-Case Analysis of Algorithms , chapter Nearest Neighbor Classification and Search
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Sparse Navigable Graphs for Nearest Neighbor Search: Algorithms and Hardness
Sparsest navigable graph construction is approximation-equivalent to Set Cover, yielding a (ln n+1)-approximation, o(ln n)-hardness, fast bicriteria algorithms, and an Ω(n squared) query lower bound.