Single-Criteria Metric $r$-Dominating Set Problem via Minor-Preserving Support
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2026
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| author | Browne, Reilly Chang, Hsien-Chih |
| author_facet | Browne, Reilly Chang, Hsien-Chih |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_00219 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Single-Criteria Metric $r$-Dominating Set Problem via Minor-Preserving Support Browne, Reilly Chang, Hsien-Chih Data Structures and Algorithms Computational Geometry 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. |
| title | Single-Criteria Metric $r$-Dominating Set Problem via Minor-Preserving Support |
| topic | Data Structures and Algorithms Computational Geometry |
| url | https://arxiv.org/abs/2604.00219 |