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Finer-Grained Reductions in Fine-Grained Hardness of Approximation

Authors Elie Abboud , Noga Ron-Zewi

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Author Details

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

Funding

  • Abboud, Elie: Research supported in part by ISF grant 735/20, and by the European Union (ERC, ECCC, 101076663). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.
  • Ron-Zewi, Noga: Research supported in part by ISF grant 735/20, and by the European Union (ERC, ECCC, 101076663). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.

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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)
https://doi.org/10.4230/LIPIcs.ICALP.2024.7

Abstract

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
Keywords
  • 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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