TY - GEN

T1 - Finer-Grained Reductions in Fine-Grained Hardness of Approximation

AU - Abboud, Elie

AU - Ron-Zewi, Noga

N1 - Publisher Copyright:
© Elie Abboud and Noga Ron-Zewi.

PY - 2024/7/1

Y1 - 2024/7/1

N2 - We investigate the relation between δ and ϵ required for obtaining a (1 + δ)-approximation in time N2−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 N2−ϵ, then there is no (1 + δ)approximation algorithm for (bichromatic) Euclidean Closest Pair running in time N2−2ϵ, 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).

AB - We investigate the relation between δ and ϵ required for obtaining a (1 + δ)-approximation in time N2−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 N2−ϵ, then there is no (1 + δ)approximation algorithm for (bichromatic) Euclidean Closest Pair running in time N2−2ϵ, 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).

KW - Analysis of algorithms

KW - Approximation algorithms

KW - Computational

KW - Computational geometry

KW - conditional lower bound

KW - Fine-grained complexity

KW - fine-grained reduction

KW - structural complexity theory

UR - http://www.scopus.com/inward/record.url?scp=85198326674&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ICALP.2024.7

DO - 10.4230/LIPIcs.ICALP.2024.7

M3 - Conference contribution

AN - SCOPUS:85198326674

T3 - Leibniz International Proceedings in Informatics, LIPIcs

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

A2 - Bringmann, Karl

A2 - Grohe, Martin

A2 - Puppis, Gabriele

A2 - Svensson, Ola

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 51st International Colloquium on Automata, Languages, and Programming, ICALP 2024

Y2 - 8 July 2024 through 12 July 2024

ER -