A tight local algorithm for the minimum dominating set problem in outerplanar graphs

TL;DR

This paper presents a deterministic local algorithm that efficiently approximates the minimum dominating set in outerplanar graphs within a factor of 5, using minimal information and no large messages or IDs.

Contribution

It introduces a constant-time distributed algorithm achieving a 5-approximation for outerplanar graphs and proves the impossibility of better approximations with similar constraints.

Findings

The algorithm guarantees a 5-approximation.

No deterministic local algorithm can surpass a 5-approximation.

The algorithm operates with minimal knowledge, only requiring vertex degrees and neighbors.

Abstract

We show that there is a deterministic local algorithm (constant-time distributed graph algorithm) that finds a 5-approximation of a minimum dominating set on outerplanar graphs. We show there is no such algorithm that finds a $(5-\varepsilon)$-approximation, for any $\varepsilon>0$. Our algorithm only requires knowledge of the degree of a vertex and of its neighbors, so that large messages and unique identifiers are not needed.