Single-Criteria Metric $r$-Dominating Set Problem via Minor-Preserving Support

Abstract: 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.