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arXiv · 2305.01892

On the Fine-Grained Complexity of Small-Size Geometric Set Cover and Discrete $k$-Center for Small $k$

Timothy M. Chan·Qizheng He·Yuancheng Yu
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Abstract

We study the time complexity of the discrete $k$-center 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 3-center in 2D, running in $\widetilde{O}(n^{3/2})$ time. - We prove a lower bound of $Ω(n^{4/3-δ})$ for rectilinear discrete 3-center in 4D, for any constant $δ>0$, under a standard hypothesis about triangle detection in sparse graphs. - Given $n$ points and $n$ weighted axis-aligned unit squares in 2D, we give the first subquadratic algorithm for finding a minimum-weight cover of the points by 3 unit squares, running in $\widetilde{O}(n^{8/5})$ time. We also prove a lower bound of $Ω(n^{3/2-δ})$ for the same problem in 2D, under the well-known APSP Hypothesis. For arbitrary axis-aligned rectangles in 2D, our upper bound is $\widetilde{O}(n^{7/4})$. - We prove a lower bound of $Ω(n^{2-δ})$ for Euclidean discrete 2-center in 13D, under the Hyperclique Hypothesis. This lower bound nearly matches the straightforward upper bound of $\widetilde{O}(n^ω)$, if the matrix multiplication exponent $ω$ is equal to 2. - We similarly prove an $Ω(n^{k-δ})$ lower bound for Euclidean discrete $k$-center in $O(k)$ dimensions for any constant $k\ge 3$, under the Hyperclique Hypothesis. This lower bound again nearly matches known upper bounds if $ω=2$. - We also prove an $Ω(n^{2-δ})$ 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 near-quadratic upper bound.

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Timothy M. Chan, Qizheng He, Yuancheng Yu. 2023-05-03. On the Fine-Grained Complexity of Small-Size Geometric Set Cover and Discrete $k$-Center for Small $k$. https://doi.org/10.4230/lipics.icalp.2023.89

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