Finer-grained Reductions in Fine-grained Hardness of Approximation.

We investigate the relation between $\delta$ and $\epsilon$ required for obtaining a $(1+\delta)$-approximation in time $N^{2-\epsilon}$ for closest pair problems under various distance metrics, and for other related problems in fine-grained complexity. Specifically, our main result shows that if it is impossible to (exactly) solve the (bichromatic) inner product (IP) problem for vectors of dimension $c \log N$ in time $N^{2-\epsilon}$, then there is no $(1+\delta)$-approximation algorithm for (bichromatic) Euclidean Closest Pair running in time $N^{2-2\epsilon}$, where $\delta \approx (\epsilon/c)^2$ (where $\approx$ hides $\polylog$ factors). This improves on the prior result due to Chen and Williams (SODA 2019) which gave a smaller polynomial dependence of $\delta$ on $\epsilon$, on the order of $\delta \approx (\epsilon/c)^6$. Our result implies in turn that no $(1+\delta)$-approximation algorithm exists for Euclidean closest pair for $\delta \approx \epsilon^4$, unless an algorithmic improvement for IP is obtained. This in turn is very close to the approximation guarantee of $\delta \approx \epsilon^3$ for Euclidean closest pair, given by the best known algorithm of Almam, Chan, and Williams (FOCS 2016). By known reductions, a similar result follows for a host of other related problems in fine-grained hardness of approximation. Our reduction combines the hardness of approximation framework of Chen and Williams, together with an MA communication protocol for IP over a small alphabet, that is inspired by the MA protocol of Chen (Theory of Computing, 2020).