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Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2022.28
URN: urn:nbn:de:0030-drops-160365
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16036/
Chitnis, Rajesh ;
Saurabh, Nitin
Tight Lower Bounds for Approximate & Exact k-Center in ℝ^d
Abstract
In the discrete k-Center problem, we are given a metric space (P,dist) where |P| = n and the goal is to select a set C ⊆ P of k centers which minimizes the maximum distance of a point in P from its nearest center. For any ε > 0, Agarwal and Procopiuc [SODA '98, Algorithmica '02] designed an (1+ε)-approximation algorithm for this problem in d-dimensional Euclidean space which runs in O(dn log k) + (k/ε)^{O (k^{1-1/d})}⋅ n^{O(1)} time. In this paper we show that their algorithm is essentially optimal: if for some d ≥ 2 and some computable function f, there is an f(k)⋅(1/ε)^{o (k^{1-1/d})} ⋅ n^{o (k^{1-1/d})} time algorithm for (1+ε)-approximating the discrete k-Center on n points in d-dimensional Euclidean space then the Exponential Time Hypothesis (ETH) fails.
We obtain our lower bound by designing a gap reduction from a d-dimensional constraint satisfaction problem (CSP) to discrete d-dimensional k-Center. This reduction has the property that there is a fixed value ε (depending on the CSP) such that the optimal radius of k-Center instances corresponding to satisfiable and unsatisfiable instances of the CSP is < 1 and ≥ (1+ε) respectively. Our claimed lower bound on the running time for approximating discrete k-Center in d-dimensions then follows from the lower bound due to Marx and Sidiropoulos [SoCG '14] for checking the satisfiability of the aforementioned d-dimensional CSP.
As a byproduct of our reduction, we also obtain that the exact algorithm of Agarwal and Procopiuc [SODA '98, Algorithmica '02] which runs in n^{O (d⋅ k^{1-1/d})} time for discrete k-Center on n points in d-dimensional Euclidean space is asymptotically optimal. Formally, we show that if for some d ≥ 2 and some computable function f, there is an f(k)⋅n^{o (k^{1-1/d})} time exact algorithm for the discrete k-Center problem on n points in d-dimensional Euclidean space then the Exponential Time Hypothesis (ETH) fails. Previously, such a lower bound was only known for d = 2 and was implicit in the work of Marx [IWPEC '06].
BibTeX - Entry
@InProceedings{chitnis_et_al:LIPIcs.SoCG.2022.28,
author = {Chitnis, Rajesh and Saurabh, Nitin},
title = {{Tight Lower Bounds for Approximate \& Exact k-Center in \mathbb{R}^d}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {28:1--28:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16036},
URN = {urn:nbn:de:0030-drops-160365},
doi = {10.4230/LIPIcs.SoCG.2022.28},
annote = {Keywords: k-center, Euclidean space, Exponential Time Hypothesis (ETH), lower bound}
}
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Keywords: |
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k-center, Euclidean space, Exponential Time Hypothesis (ETH), lower bound |
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Collection: |
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38th International Symposium on Computational Geometry (SoCG 2022) |
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Issue Date: |
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2022 |
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Date of publication: |
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01.06.2022 |
