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

2026-03-31

AI Summary

This work addresses the metric $ r $-domination problem on planar graphs with vertex weights and arbitrary radius $ r $, presenting the first single-criterion polynomial-time constant-factor approximation algorithm. The approach constructs a support graph via Voronoi cell contraction to preserve sparsity and introduces a novel assignment scheme that uniquely allocates cells of depth at least three to triples of ball centers. By combining Clarkson–Shor sampling techniques with geometric arguments, the authors establish a linear bound on the complexity of shallow cells. Leveraging quasi-uniform sampling and structural properties of planar graphs, the algorithm achieves an $ O(1) $-approximation ratio without relaxing the radius constraint, thereby overcoming the prior limitation that only bicriteria approximations were known.

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.

Problem

• r-dominating set
• planar graphs
• vertex-weighted
• metric
• approximation algorithm

Innovation

• single-criteria approximation
• metric r-dominating set
• planar graphs
• shallow cell complexity
• Voronoi contraction