Given a set of $n$ points in $\mathbb R^d$, the (monochromatic) Closest Pair problem asks to find a pair of distinct points in the set that are closest in the $\ell_p$-metric. Closest Pair is a fundamental problem in Computational Geometry and understanding its fine-grained complexity in the Euclidean metric when $d=\omega(\log n)$ was raised as an open question in recent works (Abboud-Rubinstein-Williams [FOCS'17], Williams [SODA'18], David-Karthik-Laekhanukit [SoCG'18]).
In this paper, we show that for every $p\in\mathbb R_{\ge 1}\cup\{0\}$, under the Strong Exponential Time Hypothesis (SETH), for every $\varepsilon>0$, the following holds:
$\bullet$ No algorithm running in time $O(n^{2-\varepsilon})$ can solve the Closest Pair problem in $d=(\log n)^{\Omega_{\varepsilon}(1)}$ dimensions in the $\ell_p$-metric.
$\bullet$ There exists $\delta = \delta(\varepsilon)>0$ and $c = c(\varepsilon)\ge 1$ such that no algorithm running in time $O(n^{1.5-\varepsilon})$ can approximate Closest Pair problem to a factor of $(1+\delta)$ in $d\ge c\log n$ dimensions in the $\ell_p$-metric.
In particular, our first result is shown by establishing the computational equivalence of the bichromatic Closest Pair problem and the (monochromatic) Closest Pair problem (up to $n^{\varepsilon}$ factor in the running time) for $d=(\log n)^{\Omega_\varepsilon(1)}$ dimensions.
Additionally, under SETH, we rule out nearly-polynomial factor approximation algorithms running in subquadratic time for the (monochromatic) Maximum Inner Product problem where we are given a set of $n$ points in $n^{o(1)}$-dimensional Euclidean space and are required to find a pair of distinct points in the set that maximize the inner product.
At the heart of all our proofs is the construction of a dense bipartite graph with low contact dimension, i.e., we construct a balanced bipartite graph on $n$ vertices with $n^{2-\varepsilon}$ edges whose vertices can be realized as points in a $(\log n)^{\Omega_\varepsilon(1)}$-dimensional Euclidean space such that every pair of vertices which have an edge in the graph are at distance exactly 1 and every other pair of vertices are at distance greater than 1. This graph construction is inspired by the construction of locally dense codes introduced by Dumer-Miccancio-Sudan [IEEE Trans. Inf. Theory'03].
