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 - Fine-grained complexity
KW - conditional lower bound
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 -