Abstract
Given a set of n points in ℝ^{d}, the (monochromatic) Closest Pair problem asks to find a pair of distinct points in the set that are closest in the ℓ_{p}metric. Closest Pair is a fundamental problem in Computational Geometry and understanding its finegrained complexity in the Euclidean metric when d = ω(log n) was raised as an open question in recent works (AbboudRubinsteinWilliams [6], Williams [48], DavidKarthikLaekhanukit [17]).
In this paper, we show that for every p ∈ ℝ_{≥1} ∪ {0}, under the Strong Exponential Time Hypothesis (SETH), for every ε > 0, the following holds:

No algorithm running in time O(n^{2−ε}) can solve the Closest Pair problem in \(d = {\left({\log n} \right)^{{\Omega _\varepsilon}\left(1 \right)}}\) dimensions in the ℓ_{p}metric.

There exists δ = δ(ε) > 0 and c = c(ε) ≥ 1 such that no algorithm running in time O(n^{1.5−ε}) can approximate Closest Pair problem to a factor of (1 + δ) in d ≥ c log n dimensions in the ℓ_{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^{ε} factor in the running time) for \(d = {\left({\log n} \right)^{{\Omega _\varepsilon}\left(1 \right)}}\) dimensions.
Additionally, under SETH, we rule out nearlypolynomial 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−ε} edges whose vertices can be realized as points in a \({\left({\log n} \right)^{{\Omega _\varepsilon}\left(1 \right)}}\)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 DumerMicciancioSudan [18].
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Acknowledgements
We are grateful to Madhu Sudan for extremely helpful and informative discussion about AG codes; in particular, Madhu pointed us to [46]. We thank Bundit Laekhanukit and Or Meir for general discussions, and the Simons Institute for their wonderful workspace. Finally, we would like to thank Lijie Chen and Orr Paradise for useful comments on the paper.
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Full version of the paper available at http://arxiv.org/abs/1812.00901
Supported by Irit Dinur’s ERCCoG grant 772839 and BSF grant 2014371.
Supported by NSF under Grants No. CCF 1655215 and CCF 1815434.
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Karthik, C.S., Manurangsi, P. On Closest Pair in Euclidean Metric: Monochromatic is as Hard as Bichromatic. Combinatorica 40, 539–573 (2020). https://doi.org/10.1007/s0049301941131
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Mathematics Subject Classification (2010)
 68Q17
 68R10
 05C62
 52C17