The complexity of LSH feasibility
In this paper we study the complexity of the following feasibility problem: given an nn n  n similarity matrix S as input, is there a locality sensitive hash (LSH) for S? We show that the LSH feasibility problem is NP-hard even in the following strong promise version: either S admits an LSH or S  is at ?1 ? 1 -distance at least n2?? n 2 ? ? from every similarity that admits an LSH. We complement this hardness result by providing an O  ( 3 n ) algorithm for the LSH feasibility problem, which improves upon the nave n?(n) n ? ( n ) time algorithm; we prove that this running time is tight, modulo constants, under the Exponential Time Hypothesis.
lsh linear duality