发表机构
National Institute of Science Education and Research; Homi Bhabha National Institute; IIIT-Delhi; NYU Abu Dhabi（印度科学教育研究所; 霍米·巴巴国立大学; 德里信息技术研究所; 纽约大学阿布扎比分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对几何区域的打包与覆盖问题，以穿刺度为参数，证明固定穿刺度下标准局部搜索算法可实现PTAS，并扩展至加权情形的确定性近似保证。
AI 中文摘要
几何区域的打包与覆盖问题已在多种复杂度概念下被研究，包括VC维、并复杂度、浅单元复杂度和胖度。尽管这些限制通常能产生常数因子近似算法，但它们本身一般不会导致PTAS。已知硬度构造的一个反复出现的特征是，一个区域可能被许多其他区域“穿刺”：当A\B不连通时，区域B穿刺A。我们通过“穿刺度”来研究几何实例。由于穿刺对于Jordan区域是对称的，这对应于相应穿刺图的最大度。我们的主要结果是，对于每个固定的穿刺度，标准局部搜索算法为无权“离散独立集”和“集合覆盖”问题提供PTAS。证明为适当的局部性图构造了一个亚线性平衡分隔符，并在递归局部搜索分析中自适应地应用它。该保证仅依赖于穿刺度；特别是，它不对单个穿刺对所创建的组件数量施加限制。我们还证明了一个仅依赖于穿刺度的多项式浅迹界。作为推论，对于每个固定的穿刺度，加权集合覆盖允许确定性C_r近似，加权离散独立集允许确定性O(r+1)近似。这些结果扩展了非穿刺族的已知保证，并适用于例如轴平行矩形，当每个矩形仅被有限数量的其他矩形穿刺时。
英文摘要
Packing and covering problems for geometric regions have been studied under many notions of complexity, including VC-dimension, union complexity, shallow-cell complexity, and fatness. Although these restrictions often yield constant-factor approximation algorithms, they do not by themselves generally lead to PTASs. A recurring feature of known hardness constructions is that one region may be \emph{pierced} by many others: a region $B$ pierces $A$ when $A\setminus B$ is disconnected. We study geometric instances through the \emph{piercing degree}. Since piercing is symmetric for Jordan regions, this is the maximum degree of the corresponding piercing graph. Our main result is that, for every fixed piercing degree, the standard local-search algorithms give PTASs for the unweighted \emph{Discrete Independent Set} and \emph{Set Cover} problems. The proof constructs a sublinear balanced separator for an appropriate locality graph and applies it adaptively throughout the recursive local-search analysis. This guarantee depends only on the piercing degree; in particular, it places no bound on the number of components created by an individual piercing pair. We also prove a polynomial shallow-trace bound depending only on the piercing degree. As consequences, for every fixed piercing degree, weighted Set Cover admits a deterministic $C_r$-approximation and weighted Discrete Independent Set admits a deterministic $O(r+1)$-approximation. These results extend the known guarantees for non-piercing families and apply, for example, to axis-parallel rectangles when every rectangle is pierced by only a bounded number of other rectangles.
CommentsThis is a corrected version of our ICALP 2026 paper. Unforunately, the conference version has an unfixable bug rendering the proofs incorrect. The current version proves the stronger (albeit with slightly worse running time)-the conference version required also a bound on the fragmentation number. This version removes the dependence on the fragmentation number
