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).

翻译：我们研究了在不同距离度量下最近点对问题以及细粒度复杂性中其他相关问题中，为在时间$N^{2-\epsilon}$内获得$(1+\delta)$-近似所需满足的$\delta$与$\epsilon$之间的关系。具体来说，我们的主要结果表明：若对于维数为$c\log N$的向量（双色）内积问题，无法在时间$N^{2-\epsilon}$内（精确）求解，则对于（双色）欧几里得最近点对问题，不存在运行时间为$N^{2-2\epsilon}$的$(1+\delta)$-近似算法，其中$\delta \approx (\epsilon/c)^2$（$\approx$隐含$\polylog$因子）。这一结果改进了Chen与Williams（SODA 2019）的先前结论，其给出的$\delta$关于$\epsilon$的依赖关系为更小的多项式阶次$\delta \approx (\epsilon/c)^6$。我们的结果进而表明：除非内积问题的算法性能得到改进，否则对于欧几里得最近点对问题，不存在$\delta \approx \epsilon^4$的$(1+\delta)$-近似算法。这一数值非常接近Almam、Chan与Williams（FOCS 2016）已知最优算法给出的近似保证$\delta \approx \epsilon^3$。通过已知归约，类似结论可推广至细粒度近似硬度中的一系列其他相关问题。我们的归约融合了Chen与Williams的近似硬度框架，以及受Chen（Theory of Computing, 2020）的MA协议启发而设计的小字母表上的内积问题MA通信协议。