Given an unweighted graph $G$, the *minimum $r$-dominating set problem* asks for the smallest-cardinality subset $S$ such that every vertex in $G$ is within radius $r$ of some vertex in $S$. While the $r$-dominating set problem on planar graphs admits a PTAS from Baker's shifting/layering technique when $r$ is constant, it becomes significantly harder when $r$ can depend on $n$. Under the Exponential-Time Hypothesis, Fox-Epstein et al. [SODA 2019] showed that no efficient PTAS exists for the unbounded $r$-dominating set problem on planar graphs. One may also consider the harder *vertex-weighted metric $r$-dominating set*, where edges have lengths, vertices have positive weights, and the goal is to find an $r$-dominating set of minimum total weight. This led to the development of *bicriteria* algorithms that allow radius-$(1+\varepsilon)r$ balls while achieving a $1+\varepsilon$ approximation to the optimal weight. We establish the first *single-criteria* polynomial-time $O(1)$-approximation algorithm for the vertex-weighted metric $r$-dominating set on planar graphs, where $r$ is part of the input and can be arbitrarily large. Our algorithm applies the quasi-uniformity sampling of Chan et al. [SODA 2012] by bounding the *shallow cell complexity* of the radius-$r$ ball system to be linear in $n$. Two technical innovations enable this: 1. Since discrete ball systems on planar graphs are neither pseudodisks nor amenable to standard union-complexity arguments, we construct a *support graph* for arbitrary distance ball systems as contractions of Voronoi cells, with sparseness as a byproduct. 2. We assign each depth-($\geq 3$) cell to a unique 3-tuple of ball centers, enabling Clarkson-Shor techniques to reduce counting to depth-*exactly*-3 cells, which we prove are $O(n)$ by a geometric argument on our Voronoi contraction support.

翻译：给定一个无权重图 $G$，*最小 $r$-支配集问题* 要求找出基数最小的子集 $S$，使得 $G$ 中每个顶点到 $S$ 中某个顶点的半径不超过 $r$。当 $r$ 为常数时，平面图上的 $r$-支配集问题可通过 Baker 的平移/分层技术获得 PTAS；但当 $r$ 可依赖于 $n$ 时，问题难度显著增加。在指数时间假设下，Fox-Epstein 等人 [SODA 2019] 证明平面图上的无界 $r$-支配集问题不存在有效 PTAS。进一步可考虑更难的*顶点加权度量 $r$-支配集*问题：边具有长度，顶点具有正权重，目标是寻找总权重最小的 $r$-支配集。这促使了*双准则*算法的发展，该类算法允许使用半径为 $(1+\varepsilon)r$ 的球体，同时将最优权重的近似比控制在 $1+\varepsilon$ 内。本文首次提出平面图上顶点加权度量 $r$-支配集问题的*单准则*多项式时间 $O(1)$-近似算法，其中 $r$ 为输入参数且可任意大。该算法应用了 Chan 等人 [SODA 2012] 的拟均匀采样方法，通过将半径-$r$ 球系统的*浅层单元复杂度*控制在 $n$ 的线性阶内。两项技术创新实现了这一突破：1. 由于平面图上的离散球系统既非伪圆盘族，亦不适用标准并复杂度论证，我们通过 Voronoi 胞腔的收缩构造了任意距离球系统的*支撑图*，并由此获得稀疏性；2. 我们将每个深度 $\geq 3$ 的单元唯一映射至球心三元组，从而利用 Clarkson-Shor 技术将计数归约至*精确深度*-3 的单元，并通过基于 Voronoi 收缩支撑的几何论证证明此类单元的数量为 $O(n)$。