A local constant factor approximation for the minimum dominating set problem on bounded genus graphs

TL;DR

This paper introduces a deterministic, local constant-factor approximation algorithm for the Minimum Dominating Set problem specifically on bounded genus graphs, expanding the class of graphs where efficient local solutions are feasible.

Contribution

It provides the first deterministic local constant-factor approximation for MDS on bounded genus graphs, using a novel analysis that avoids topological arguments.

Findings

Achieves a constant-factor approximation for MDS on bounded genus graphs.

Introduces a new analysis technique for local algorithms on sparse graphs.

Extends understanding of local algorithms beyond planar graphs.

Abstract

The Minimum Dominating Set (MDS) problem is not only one of the most fundamental problems in distributed computing, it is also one of the most challenging ones. While it is well-known that minimum dominating sets cannot be approximated locally on general graphs, over the last years, several breakthroughs have been made on computing local approximations on sparse graphs. This paper presents a deterministic and local constant factor approximation for minimum dominating sets on bounded genus graphs, a very large family of sparse graphs. Our main technical contribution is a new analysis of a slightly modified, first-order definable variant of an existing algorithm by Lenzen et al. Interestingly, unlike existing proofs for planar graphs, our analysis does not rely on any topological arguments. We believe that our techniques can be useful for the study of local problems on sparse graphs beyond the scope of this paper.