## Abstract

We describe a simple deterministic O(ε^{- 1}log Δ) round distributed algorithm for (2 α+ 1) (1 + ε) approximation of minimum weighted dominating set on graphs with arboricity at most α . Here Δ denotes the maximum degree. We also show a lower bound proving that this round complexity is nearly optimal even for the unweighted case, via a reduction from the celebrated KMW lower bound on distributed vertex cover approximation (Kuhn et al. in JACM 63:116, 2016). Our algorithm improves on all the previous results (that work only for unweighted graphs) including a randomized O(α^{2}) approximation in O(log n) rounds (Lenzen et al. in International symposium on distributed computing, Springer, 2010), a deterministic O(αlog Δ) approximation in O(log Δ) rounds (Lenzen et al. in international symposium on distributed computing, Springer, 2010), a deterministic O(α) approximation in O(log ^{2}Δ) rounds (implicit in Bansal et al. in Inform Process Lett 122:21–24, 2017; Proceeding 17th symposium on discrete algorithms (SODA), 2006), and a randomized O(α) approximation in O(αlog n) rounds (Morgan et al. in 35th International symposiumon distributed computing, 2021). We also provide a randomized O(αlog Δ) round distributed algorithm that sharpens the approximation factor to α(1 + o(1)) . If each node is restricted to do polynomial-time computations, our approximation factor is tight in the first order as it is NP-hard to achieve α- 1 - ε approximation (Bansal et al. in Inform Process Lett 122:21-24, 2017).

Original language | English |
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Journal | Distributed Computing |

DOIs | |

State | Accepted/In press - 2023 |

### Bibliographical note

Funding Information:This work was supported in part by funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 853109), and the Swiss National Foundation (project grant 200021-184735).

Publisher Copyright:

© 2023, The Author(s).

## Keywords

- Approximation algorithms
- Arboricity
- Distributed computing
- Dominating set

## ASJC Scopus subject areas

- Theoretical Computer Science
- Hardware and Architecture
- Computer Networks and Communications
- Computational Theory and Mathematics