小尺寸几何集合覆盖与小$k$值离散$k$-中心的细粒度复杂度研究
On the Fine-Grained Complexity of Small-Size Geometric Set Cover and Discrete $k$-Center for Small $k$
AI总结：
本文研究小$k$值下离散$k$-中心与小尺寸几何集合覆盖的细粒度复杂度，提出多个次平方算法并基于主流假设证明多维度、多场景下的问题下界，部分下界与已知上界几乎匹配。
摘要：
我们研究了当$k$或覆盖规模较小时，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度，得到了一系列新结果：\n- 我们提出了首个针对二维曼哈顿距离（rectilinear）离散3-中心的次平方算法，时间复杂度为$\tilde{O}(n^{3/2})$。\n- 基于稀疏图中三角形检测的标准假设，我们证明了四维曼哈顿距离离散3-中心的下界为$\tilde{\n}^{4/3-\n}$，其中$\n$为任意正常数。\n- 针对二维平面中$n$个点和$n$个带权轴对齐单位正方形的场景，我们提出了首个用3个单位正方形寻找点集最小权覆盖的次平方算法，时间复杂度为$\tilde{O}(n^{8/5})$。同时基于知名的APSP（全源最短路径）假设，证明了该二维问题的下界为$\tilde{\n}^{3/2-\n}$。对于二维中任意轴对齐矩形的情况，我们的上界为$\tilde{O}(n^{7/4})$。\n- 基于超团（Hyperclique）假设，我们证明了13维欧氏离散2-中心的下界为$\tilde{\n}^{2-\n}$。若矩阵乘法指数$\n$等于2，该下界与直观的$\tilde{O}(n^\n)$上界几乎匹配。\n- 类似地，基于超团假设，我们证明了对于任意常数$k\neq3$，$O(k)$维空间中欧氏离散$k$-中心的下界为$\tilde{\n}^{k-\n}$。若$\n=2$，该下界同样与已知上界几乎匹配。\n- 我们还证明了在12维空间中，给定$n$个点和$n$个盒子，寻找2个盒子覆盖最多点的问题的下界为$\tilde{\n}^{2-\n}$，这与直观的近二次上界相匹配。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
