几何最大覆盖的近似算法
Approximation Algorithms for Geometric Maximum Coverage
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中文总结 AI 辅助
本文研究几何最大覆盖问题，提出多种近似算法，针对不同几何对象族改进了近似因子与运行时间，并给出了对应的难度下界结果。
中文摘要 AI 辅助
我们研究几何集合系统的最大覆盖问题：给定一组点、一组几何对象以及一个数$k$，选取$k$个对象，使得其并集内部包含的点的数量最大化。\n- 我们提出了一种多项式时间近似算法，对于任何具有线性2-浅元胞复杂度的集合系统（或任何可分解为常数个此类集合系统的集合系统），其近似因子严格优于$1-1/e$。该结果同样适用于加权最大覆盖问题——在该问题中，对象具有权重，我们需要选取总权重在给定预算内的对象。该结果适用于多种类型的几何对象，包括二维伪圆盘、二维胖轴对齐矩形、二维相似尺寸胖三角形、三维轴对齐单位立方体。\n- 对于较小的$k$，我们更普遍地得到了一种适用于任何具有常数VC维的集合系统的$(1-ε)$-近似算法，其运行时间的指数项为$\tilde{O}(k/ε)$。这简化并改进了Badanidiyuru、Kleinberg和Lee提出的参数化近似方案[SoCG'12]，该方案的运行时间指数项为$\tilde{O}(k^2/ε^5)$。\n- 几何最大覆盖问题的一个连续版本要求选取$k$个对象，使其并集的体积最大化。我们针对特定对象族给出了该问题的更优近似算法；例如，我们得到了适用于任意常数维胖凸对象的EPTAS（高效多项式时间近似方案）。\n- 我们通过若干难度结果补充了我们的算法，例如二维胖轴对齐矩形的APX难度、在依赖于$ε$的维度下轴对齐盒的$(1-1/e+ε)$-近似难度，以及三维轴对齐盒的连续问题排除$n^{\mathop{\rm poly}(1/ε)}$时间PTAS（多项式时间近似方案）的下界。
英文摘要
We study the maximum coverage problem for geometric set systems: given a set of points, a set of geometric objects, and a number $k$, select $k$ objects maximizing the number of points inside their union. - We present a polynomial-time approximation algorithm with approximation factor strictly better than $1-1/e$ for any set system with linear 2-shallow cell complexity (or any set system that can be decomposed into a constant number of such set systems). The result also holds for the weighted maximum coverage problem, where objects have weights and we want to select objects with total weight within a given budget. The result applies to many types of geometric objects, including pseudodisks in 2D, fat axis-aligned rectangles in 2D, similar-size fat triangles in 2D, axis-aligned unit cubes in 3D. - For small $k$, we obtain a $(1-ε)$-approximation algorithm more generally for any set system with constant VC dimension, running in time exponential in $\tilde{O}(k/ε)$. This simplifies and improves Badanidiyuru, Kleinberg, and Lee's parameterized approximation scheme [SoCG'12] running in time exponential in $\tilde{O}(k^2/ε^5)$. - A continuous version of the geometric maximum coverage problem asks for $k$ objects maximizing the volume of their union. We give better approximation algorithms for this problem for certain families of objects; e.g., we obtain an EPTAS for fat convex objects in any constant dimension. - We complement our algorithms with several hardness results, e.g., APX-hardness for fat axis-aligned rectangles in 2D, $(1-1/e+ε)$-approximation hardness for axis-aligned boxes in a dimension dependent on $ε$, and a lower bound ruling out $n^{\mathop{\rm poly}(1/ε)}$-time PTASs for the continuous problem for axis-aligned boxes in 3D.
