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

翻译：我们研究了当$k$或覆盖大小较小时，离散$k$-中心问题及相关（精确）几何集合覆盖问题的时间复杂性。我们获得了一系列新结果： - 我们首次提出了二维矩形离散3-中心的次二次算法，运行时间为$\widetilde{O}(n^{3/2})$。 - 在稀疏图三角形检测的标准假设下，我们证明了四维矩形离散3-中心的下界为$\Omega(n^{4/3-\delta})$，其中$\delta>0$为任意常数。 - 给定二维平面上的$n$个点与$n$个带权轴对齐单位正方形，我们首次提出了用3个单位正方形覆盖点的最小权重覆盖的次二次算法，运行时间为$\widetilde{O}(n^{8/5})$。同时，在著名的APSP假设下，我们证明了该问题在二维中具有$\Omega(n^{3/2-\delta})$的下界。对于二维中任意轴对齐矩形，我们的上界为$\widetilde{O}(n^{7/4})$。 - 在超团假设下，我们证明了13维欧氏离散2-中心的下界为$\Omega(n^{2-\delta})$。若矩阵乘法指数$\omega=2$，此下界几乎匹配平凡上界$\widetilde{O}(n^\omega)$。 - 类似地，在超团假设下，我们证明了对于任意常数$k\ge 3$，$O(k)$维欧氏离散$k$-中心的下界为$\Omega(n^{k-\delta})$。当$\omega=2$时，此下界再次几乎匹配已知上界。 - 我们还证明了给定12维空间中的$n$个点与$n$个盒时，用2个盒覆盖最多点数的问题的下界为$\Omega(n^{2-\delta})$，这匹配了平凡的近二次上界。