The complexity of LSH feasibility

In this paper we study the complexity of the following feasibility problem: given an 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 l1-distance at least n2−ϵ from every similarity that admits an LSH. We complement this hardness result by providing an O˜(3n) algorithm for the LSH feasibility problem, which improves upon the naive nΘ(n) time algorithm; we prove that this running time is tight, modulo constants, under the Exponential Time Hypothesis.

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