## 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 ≈ ϵ^{4}, unless an algorithmic improvement for IP is obtained. This in turn is very close to the approximation guarantee of δ ≈ ϵ^{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).

Original language | English |
---|---|

Title of host publication | 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024 |

Editors | Karl Bringmann, Martin Grohe, Gabriele Puppis, Ola Svensson |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959773225 |

DOIs | |

State | Published - Jul 2024 |

Event | 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024 - Tallinn, Estonia Duration: 8 Jul 2024 → 12 Jul 2024 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
---|---|

Volume | 297 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024 |
---|---|

Country/Territory | Estonia |

City | Tallinn |

Period | 8/07/24 → 12/07/24 |

### Bibliographical note

Publisher Copyright:© Elie Abboud and Noga Ron-Zewi.

## Keywords

- Analysis of algorithms
- Approximation algorithms
- Computational
- Computational geometry
- Fine-grained complexity
- conditional lower bound
- fine-grained reduction
- structural complexity theory

## ASJC Scopus subject areas

- Software