Hardness of Bichromatic Closest Pair with Jaccard Similarity

Abstract

Consider collections $\mathcal{A}$ and $\mathcal{B}$ of red and blue sets, respectively. Bichromatic Closest Pair is the problem of finding a pair from $\mathcal{A}\times \mathcal{B}$ that has similarity higher than a given threshold according to some similarity measure. Our focus here is the classic Jaccard similarity $|\textbf{a}\cap \textbf{b}|/|\textbf{a}\cup \textbf{b}|$ for $(\textbf{a},\textbf{b})\in \mathcal{A}\times \mathcal{B}$. We consider the approximate version of the problem where we are given thresholds $j_1>j_2$ and wish to return a pair from $\mathcal{A}\times \mathcal{B}$ that has Jaccard similarity higher than $j_2$ if there exists a pair in $\mathcal{A}\times \mathcal{B}$ with Jaccard similarity at least $j_1$. The classic locality sensitive hashing (LSH) algorithm of Indyk and Motwani (STOC '98), instantiated with the MinHash LSH function of Broder et al., solves this problem in $\tilde O(n^{2-δ})$ time if $j_1\ge j_2^{1-δ}$. In particular, for $δ=Ω(1)$, the approximation ratio $j_1/j_2=1/j_2^δ$ increases polynomially in $1/j_2$. In this paper we give a corresponding hardness result. Assuming the Orthogonal Vectors Conjecture (OVC), we show that there cannot be a general solution that solves the Bichromatic Closest Pair problem in $O(n^{2-Ω(1)})$ time for $j_1/j_2=1/j_2^{o(1)}$. Specifically, assuming OVC, we prove that for any $δ>0$ there exists an $\varepsilon>0$ such that Bichromatic Closest Pair with Jaccard similarity requires time $Ω(n^{2-δ})$ for any choice of thresholds $j_2<j_1<1-δ$, that satisfy $j_1\le j_2^{1-\varepsilon}$.