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

Reilly Browne, Hsien-Chih Chang

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.

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

Abstract

Given an unweighted graph , the *minimum -dominating set problem* asks for the smallest-cardinality subset such that every vertex in is within radius of some vertex in . While the -dominating set problem on planar graphs admits a PTAS from Baker's shifting/layering technique when is constant, it becomes significantly harder when can depend on . Under the Exponential-Time Hypothesis, Fox-Epstein et al. [SODA 2019] showed that no efficient PTAS exists for the unbounded -dominating set problem on planar graphs. One may also consider the harder *vertex-weighted metric -dominating set*, where edges have lengths, vertices have positive weights, and the goal is to find an -dominating set of minimum total weight. This led to the development of *bicriteria* algorithms that allow radius- balls while achieving a approximation to the optimal weight. We establish the first *single-criteria* polynomial-time -approximation algorithm for the vertex-weighted metric -dominating set on planar graphs, where 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- ball system to be linear in . 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-() 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 by a geometric argument on our Voronoi contraction support.

Paper Structure

This paper contains 19 sections, 14 theorems, 7 figures.

Table of Contents

  1. Introduction
  2. The role of distance balls.
  3. Arbitrary radii and arbitrary weights.
  4. Our contribution.
  5. Techniques
  6. Challenge 1: Planar support.
  7. Challenge 2: Linear shallow cell complexity.
  8. Terminology and Support Graphs
  9. Applications
  10. Related Work on Support Graphs
  11. Existence of Support Graph
  12. Shallow Cell Complexity of Balls in Planar Graphs
  13. Unique 3-Site Encoding of Cells
  14. Depth-3 Cells
  15. Missing proofs
  16. ...and 4 more sections

Key Result

Theorem 1

Let $G$ be an arbitrary $n$-node planar graph, where each vertex $v$ is associated with a weight $w(v)$. Let $r$ be an arbitrary radius parameter. There is a polynomial-time algorithm that computes an $r$-dominating set in $G$ whose weight is $O(1)$ times the optimum. $\blacktriangleleft$$\blacktria

Figures (7)

  • Figure 1: Two distance balls $R$ and $R'$ that are piercing. Here both $R$ and $R'$ have radius 6.
  • Figure 2: A set system with a dual support. The red regions containing the violet vertex $v$ form a connected subgraph in the support.
  • Figure 3: An illustration of the proof to Lemma \ref{['lma:dual_is_a_support']} that $\pi$ can only pass through Voronoi cells of $\mathcal{R}_v$. The purple region is the Voronoi cell $F_Q$
  • Figure 4: Given two vertices $v,v'$ satisfying $\alpha(v) = \alpha(v') = D_1$, $\beta(v) = \beta(v') = D_2$ and $v' \in \pi(D_1,v)$, any ball $D_3$ containing $v'$ must also contain $v$.
  • Figure 6: Showing the subwalk of Lemma \ref{['lma:subwalk_new']} and how to select $\mathit{base}$. Left: Case 1. Right: Case 2.
  • ...and 2 more figures

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2: Support
  • Theorem 3: Shallow Cell Complexity
  • Lemma 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Lemma 9
  • Lemma 10
  • Lemma 11
  • ...and 4 more