# Deterministic Distributed Dominating Set Approximation in the CONGEST Model

## TL;DR

This paper presents new deterministic algorithms for approximating the minimum dominating set and connected dominating set problems in the CONGEST distributed model, achieving near-optimal guarantees with efficient runtimes.

## Contribution

It introduces deterministic approximation algorithms with improved guarantees and runtimes for dominating set and connected dominating set problems in the CONGEST model.

## Key findings

- Achieves near-optimal approximation guarantees for dominating sets.
- Provides efficient deterministic algorithms with subexponential runtimes.
- Transforms dominating set approximations into connected dominating sets with constant factor increase.

## Abstract

We develop deterministic approximation algorithms for the minimum dominating set problem in the CONGEST model with an almost optimal approximation guarantee. For $\epsilon>1/{\text{{poly}}}\log \Delta$ we obtain two algorithms with approximation factor $(1+\epsilon)(1+\ln (\Delta+1))$ and with runtimes $2^{O(\sqrt{\log n \log\log n})}$ and $O(\Delta\cdot\text{poly}\log \Delta +\text{poly}\log \Delta \log^{*} n)$, respectively. Further we show how dominating set approximations can be deterministically transformed into a connected dominating set in the \CONGEST model while only increasing the approximation guarantee by a constant factor. This results in a deterministic $O(\log \Delta)$-approximation algorithm for the minimum connected dominating set with time complexity $2^{O(\sqrt{\log n \log\log n})}$.