Finer-grained reductions in fine-grained hardness of approximation

We investigate the relation between δ and ϵ required for obtaining a (1+δ)-approximation in time N^2−ϵ 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 clog⁡N in time N^2−ϵ, then there is no (1+δ)-approximation algorithm for (bichromatic) Euclidean Closest Pair running in time N^2−2ϵ, where δ≈(ϵ/c)² (where ≈ hides polylog factors). This improves on the prior result due to Chen and Williams (SODA 2019) which gave a smaller polynomial dependence of δ on ϵ, on the order of δ≈(ϵ/c)⁶. Our result implies in turn that no (1+δ)-approximation algorithm exists for Euclidean closest pair for δ≈ϵ⁴, unless an algorithmic improvement for IP is obtained. This in turn is very close to the approximation guarantee of δ≈ϵ³ 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).