Abstract
We study the time complexity of the discrete kcenter problem and related (exact) geometric set cover problems when k or the size of the cover is small. We obtain a plethora of new results:
 We give the first subquadratic algorithm for rectilinear discrete 3center in 2D, running in Õ(n^{3/2}) time.
 We prove a lower bound of Ω(n^{4/3δ}) for rectilinear discrete 3center in 4D, for any constant δ > 0, under a standard hypothesis about triangle detection in sparse graphs.
 Given n points and n weighted axisaligned unit squares in 2D, we give the first subquadratic algorithm for finding a minimumweight cover of the points by 3 unit squares, running in Õ(n^{8/5}) time. We also prove a lower bound of Ω(n^{3/2δ}) for the same problem in 2D, under the wellknown APSP Hypothesis. For arbitrary axisaligned rectangles in 2D, our upper bound is Õ(n^{7/4}).
 We prove a lower bound of Ω(n^{2δ}) for Euclidean discrete 2center in 13D, under the Hyperclique Hypothesis. This lower bound nearly matches the straightforward upper bound of Õ(n^ω), if the matrix multiplication exponent ω is equal to 2.
 We similarly prove an Ω(n^{kδ}) lower bound for Euclidean discrete kcenter in O(k) dimensions for any constant k ≥ 3, under the Hyperclique Hypothesis. This lower bound again nearly matches known upper bounds if ω = 2.
 We also prove an Ω(n^{2δ}) lower bound for the problem of finding 2 boxes to cover the largest number of points, given n points and n boxes in 12D . This matches the straightforward nearquadratic upper bound.
BibTeX  Entry
@InProceedings{chan_et_al:LIPIcs.ICALP.2023.34,
author = {Chan, Timothy M. and He, Qizheng and Yu, Yuancheng},
title = {{On the FineGrained Complexity of SmallSize Geometric Set Cover and Discrete kCenter for Small k}},
booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
pages = {34:134:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772785},
ISSN = {18688969},
year = {2023},
volume = {261},
editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/18086},
URN = {urn:nbn:de:0030drops180868},
doi = {10.4230/LIPIcs.ICALP.2023.34},
annote = {Keywords: Geometric set cover, discrete kcenter, conditional lower bounds}
}
Keywords: 

Geometric set cover, discrete kcenter, conditional lower bounds 
Collection: 

50th International Colloquium on Automata, Languages, and Programming (ICALP 2023) 
Issue Date: 

2023 
Date of publication: 

05.07.2023 