Hardness of Approximate Nearest Neighbor Search under L-infinity

We show conditional hardness of Approximate Nearest Neighbor Search (ANN) under the $\ell_\infty$ norm with two simple reductions. Our first reduction shows that hardness of a special case of the Shortest Vector Problem (SVP), which captures many provably hard instances of SVP, implies a lower bound for ANN with polynomial preprocessing time under the same norm. Combined with a recent quantitative hardness result on SVP under $\ell_\infty$ (Bennett et al., FOCS 2017), our reduction implies that finding a $(1+\varepsilon)$-approximate nearest neighbor under $\ell_\infty$ with polynomial preprocessing requires near-linear query time, unless the Strong Exponential Time Hypothesis (SETH) is false. This complements the results of Rubinstein (STOC 2018), who showed hardness of ANN under $\ell_1$, $\ell_2$, and edit distance. Further improving the approximation factor for hardness, we show that, assuming SETH, near-linear query time is required for any approximation factor less than $3$ under $\ell_\infty$. This shows a conditional separation between ANN under the $\ell_1/ \ell_2$ norm and the $\ell_\infty$ norm since there are sublinear time algorithms achieving better than $3$-approximation for the $\ell_1$ and $\ell_2$ norm. Lastly, we show that the approximation factor of $3$ is a barrier for any naive gadget reduction from the Orthogonal Vectors problem.