Finer-Grained Reductions in Fine-Grained Hardness of Approximation

Authors Elie Abboud , Noga Ron-Zewi

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Elie Abboud
  • Department of Computer Science, University of Haifa, Israel
Noga Ron-Zewi
  • Department of Computer Science, University of Haifa, Israel

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Elie Abboud and Noga Ron-Zewi. Finer-Grained Reductions in Fine-Grained Hardness of Approximation. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 7:1-7:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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 c log 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).

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Fine-grained complexity
  • conditional lower bound
  • fine-grained reduction
  • Approximation algorithms
  • Analysis of algorithms
  • Computational geometry
  • Computational and structural complexity theory


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