We initiate a systematic study of approximation schemes for fundamental optimization problems on disk graphs, a common generalization of both planar graphs and unit-disk graphs. Our main contribution is a general framework for designing efficient polynomial-time approximation schemes (EPTASes) for vertex-deletion problems on disk graphs, which results in EPTASes for many problems including Vertex Cover, Feedback Vertex Set, Small Cycle Hitting (in particular, Triangle Hitting), $P_k$-Hitting for $k\in\{3,4,5\}$, Path Deletion, Pathwidth $1$-Deletion, Component Order Connectivity, Bounded Degree Deletion, Pseudoforest Deletion, Finite-Type Component Deletion, etc. All EPTASes obtained using our framework are robust in the sense that they do not require a realization of the input graph. To the best of our knowledge, prior to this work, the only problems known to admit (E)PTASes on disk graphs are Maximum Clique, Independent Set, Dominating set, and Vertex Cover, among which the existing PTAS [Erlebach et al., SICOMP'05] and EPTAS [Leeuwen, SWAT'06] for Vertex Cover require a realization of the input disk graph (while ours does not). The core of our framework is a reduction for a broad class of (approximation) vertex-deletion problems from (general) disk graphs to disk graphs of bounded local radius, which is a new invariant of disk graphs introduced in this work. Disk graphs of bounded local radius can be viewed as a mild generalization of planar graphs, which preserves certain nice properties of planar graphs. Specifically, we prove that disk graphs of bounded local radius admit the Excluded Grid Minor property and have locally bounded treewidth. This allows existing techniques for designing approximation schemes on planar graphs (e.g., bidimensionality and Baker's technique) to be directly applied to disk graphs of bounded local radius.

翻译：我们开始系统研究磁盘图中基本优化问题的近似方案, 这是平流图和单位磁盘图的共同直径化。 我们的主要贡献是一个用于设计磁盘图中顶点偏移问题的高效多边时间近似方案( EPTASes ) 的通用框架。 这导致 EPTAS 处理许多问题, 包括 Vertex 封面、 反馈 Vertex Set、 小型循环点击( 特别是三角点击 )、 $P_ k$- 硬化 $k\ 3, 5 ⁇ $, 路径Delement、 路徑 $Delation、 组件连接、 声调调调调调调调调调调色调色调色调色调色调 。 使用我们框架获得的所有 EPTAS 都非常可靠, 它们不需要实现 输入图形。 在这项工作之前, 我们所知道的磁盘图中( E) PTEAS 直流化技术是最小的平流化, 直流调平流调平流平流的平流的平流的平流平流平流图, 直径平流的平流平流的平流平流的平流的平流的平流的平流的平流平流的平流的平流的平流的平流的平流的平流的平流化, 直径平流的平流的平流的平流的平流的平流的平流的平流的平流的平流的平流的平流的平流的平流图, 直径。