Finer-grained reductions in fine-grained hardness of approximation

Abstract

We investigate the relation between δ and ϵ required for obtaining a $(1+\delta )$-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 $c\log N$ in time $N^{2-ϵ}$, then there is no $(1+\delta )$-approximation algorithm for (bichromatic) Euclidean Closest Pair running in time $N^{2-2ϵ}$, where $\delta \approx {(ϵ/c)}^2$ (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 $\delta \approx {(ϵ/c)}^6$. Our result implies in turn that no $(1+\delta )$-approximation algorithm exists for Euclidean closest pair for $\delta \approx {ϵ}^4$, unless an algorithmic improvement for IP is obtained. This in turn is very close to the approximation guarantee of $\delta \approx {ϵ}^3$ 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).