# Distributed Dominating Set Approximations beyond Planar Graphs

## TL;DR

This paper extends distributed approximation algorithms for the Minimum Dominating Set problem from planar graphs to more complex graph classes like bounded genus graphs, introducing new algorithms with improved analysis techniques.

## Contribution

It presents the first distributed, deterministic approximation algorithms for MDS on bounded genus graphs, including a constant-factor and an approximation scheme, with novel analysis methods.

## Key findings

- Achieved a local constant-time, constant-factor MDS approximation algorithm.
- Developed an $ ext{O}( ext{log}^* n)$-time approximation scheme for MDS.
- Provided a new analysis of existing algorithms that avoids topological arguments.

## Abstract

The Minimum Dominating Set (MDS) problem is one of the most fundamental and challenging problems in distributed computing. While it is well-known that minimum dominating sets cannot be approximated locally on general graphs, over the last years, there has been much progress on computing local approximations on sparse graphs, and in particular planar graphs. In this paper we study distributed and deterministic MDS approximation algorithms for graph classes beyond planar graphs. In particular, we show that existing approximation bounds for planar graphs can be lifted to bounded genus graphs, and present (1) a local constant-time, constant-factor MDS approximation algorithm and (2) a local $\mathcal{O}(\log^*{n})$-time approximation scheme. Our main technical contribution is a new analysis of a slightly modified variant of an existing algorithm by Lenzen et al. Interestingly, unlike existing proofs for planar graphs, our analysis does not rely on direct topological arguments.