Distributed Approximation Algorithms for Minimum Dominating Set in Locally Nice Graphs

TL;DR

This paper presents improved distributed approximation algorithms for the Minimum Dominating Set problem in graphs embeddable on surfaces of bounded Euler genus, achieving better approximation ratios with fewer rounds in the LOCAL model.

Contribution

It introduces a new proof and algorithmic approach that significantly improves approximation ratios for MDS in graphs on surfaces, without requiring graph embedding preprocessing.

Findings

Achieves an approximation ratio of at most 34 + ε for MDS in graphs embeddable on surfaces.

Provides a deterministic LOCAL algorithm that works without preliminary graph embedding.

Improves upon previous approximation ratios for graphs on orientable surfaces.

Abstract

We give a new, short proof that graphs embeddable in a given Euler genus-$g$ surface admit a simple $f(g)$-round $\alpha$-approximation distributed algorithm for Minimum Dominating Set (MDS), where the approximation ratio $\alpha \le 906$. Using tricks from Heydt et al. [European Journal of Combinatorics (2025)], we in fact derive that $\alpha \le 34 +\varepsilon$, therefore improving upon the current state of the art of $24g+O(1)$ due to Amiri et al. [ACM Transactions on Algorithms (2019)]. It also improves the approximation ratio of $91+\varepsilon$ due to Czygrinow et al. [Theoretical Computer Science (2019)] in the particular case of orientable surfaces. All our distributed algorithms work in the deterministic LOCAL model. They do not require any preliminary embedding of the graph and only rely on two things: a LOCAL algorithm for MDS on planar graphs with ``uniform'' approximation guarantees and the knowledge that graphs embeddable in bounded Euler genus surfaces have asymptotic dimension $2$. More generally, our algorithms work in any graph class of bounded asymptotic dimension where ``most vertices'' are locally in a graph class that admits a LOCAL algorithm for MDS with uniform approximation guarantees.