# Distributed Domination on Graph Classes of Bounded Expansion

## TL;DR

This paper introduces a distributed approximation algorithm for the distance-$r$ dominating set problem on bounded expansion graph classes, achieving efficient computation in the ext{CONGEST}_{BC} model and improving connected dominating set approximations.

## Contribution

It presents a novel distributed algorithm for approximating distance-$r$ dominating sets on bounded expansion graphs, utilizing sparse neighborhood covers and extending to connected dominating sets.

## Key findings

- Algorithm computes $r$-neighborhood covers in $ ext{O}(r^2 ext{log} n)$ rounds.
- Provides a constant factor approximation for connected dominating sets in bounded expansion graphs.
- Achieves efficient distributed computation in the ext{CONGEST}_{BC} and ext{LOCAL} models.

## Abstract

We provide a new constant factor approximation algorithm for the (connected) distance-$r$ dominating set problem on graph classes of bounded expansion. Classes of bounded expansion include many familiar classes of sparse graphs such as planar graphs and graphs with excluded (topological) minors, and notably, these classes form the most general subgraph closed classes of graphs for which a sequential constant factor approximation algorithm for the distance-$r$ dominating set problem is currently known. Our algorithm can be implemented in the \congestbc model of distributed computing and uses $\mathcal{O}(r^2 \log n)$ communication rounds. Our techniques, which may be of independent interest, are based on a distributed computation of sparse neighborhood covers of small radius on bounded expansion classes. We show how to compute an $r$-neighborhood cover of radius~$2r$ and overlap $f(r)$ on every class of bounded expansion in $\mathcal{O}(r^2 \log n)$ communication rounds for some function~$f$.% in the $\mathcal{CONGEST}_{\mathrm{BC}}$ model. Finally, we show how to use the greater power of the $\mathcal{LOCAL}$ model to turn any distance-$r$ dominating set into a constantly larger connected distance-$r$ dominating set in $3r+1$ rounds on any class of bounded expansion. Combining this algorithm, e.g., with the constant factor approximation algorithm for dominating sets on planar graphs of Lenzen et al.\ gives a constant factor approximation algorithm for connected dominating sets on planar graphs in a constant number of rounds in the $\mathcal{LOCAL}$ model, where the approximation ratio is only $6$ times larger than that of Lenzen et al.'s algorithm.