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

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 $\tilde{O}(n^{3\mathrm{/}2})$ time. - We prove a lower bound of $\mathrm{\Omega }(n^{4\mathrm{/}3-\delta })$ for rectilinear discrete 3-center in 4D, for any constant $\delta >0$, 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 $\tilde{O}(n^{8\mathrm{/}5})$ time. We also prove a lower bound of $\mathrm{\Omega }(n^{3\mathrm{/}2-\delta })$ for the same problem in 2D, under the well-known APSP Hypothesis. For arbitrary axis-aligned rectangles in 2D, our upper bound is $\tilde{O}(n^{7\mathrm{/}4})$. - We prove a lower bound of $\mathrm{\Omega }(n^{2-\delta })$ for Euclidean discrete 2-center in 13D, under the Hyperclique Hypothesis. This lower bound nearly matches the straightforward upper bound of $\tilde{O}(n^{\omega })$, if the matrix multiplication exponent $\omega$ is equal to 2. - We similarly prove an $\mathrm{\Omega }(n^{k-\delta })$ 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 $\omega =2$. - We also prove an $\mathrm{\Omega }(n^{2-\delta })$ 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.