## Abstract

The tree augmentation problem (TAP) is a fundamental network design problem, in which the input is a graph G and a spanning tree T for it, and the goal is to augment T with a minimum set of edges Aug from G, such that T ∪ Aug is 2-edge-connected. TAP has been widely studied in the sequential setting. The best known approximation ratio of 2 for the weighted case dates back to the work of Frederickson and JáJá, SICOMP 1981. Recently, a 3/2-approximation was given for the unweighted case by Kortsarz and Nutov, TALG 2016, and recent breakthroughs by Adjiashvili, SODA 2017, and by Fiorini et al., 2017, give approximations better than 2 for bounded weights. In this paper, we provide the first fast distributed approximations for TAP. We present a distributed 2-approximation for weighted TAP which completes in O(h) rounds, where h is the height of T. When h is large, we show a much faster 4-approximation algorithm for the unweighted case, completing in O(D + √ n log∗ n) rounds, where n is the number of vertices and D is the diameter of G. Immediate consequences of our results are an O(D)-round 2-approximation algorithm for the minimum size 2-edge-connected spanning subgraph, which significantly improves upon the running time of previous approximation algorithms, and an O(h_{MST} + √ n log∗ n)-round 3-approximation algorithm for the weighted case, where h_{MST} is the height of the MST of the graph. Additional applications are algorithms for verifying 2-edge-connectivity and for augmenting the connectivity of any connected spanning subgraph to 2. Finally, we complement our study with proving lower bounds for distributed approximations of TAP.

Original language | English |
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Title of host publication | 21st International Conference on Principles of Distributed Systems, OPODIS 2017 |

Editors | James Aspnes, Joao Leitao, Alysson Bessani, Pascal Felber |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959770613 |

DOIs | |

State | Published - 1 Mar 2018 |

Externally published | Yes |

Event | 21st International Conference on Principles of Distributed Systems, OPODIS 2017 - Lisboa, Portugal Duration: 18 Dec 2017 → 20 Dec 2017 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 95 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 21st International Conference on Principles of Distributed Systems, OPODIS 2017 |
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Country/Territory | Portugal |

City | Lisboa |

Period | 18/12/17 → 20/12/17 |

### Bibliographical note

Funding Information:∗ Supported in part by the Israel Science Foundation (grant 1696/14). † The full version of the paper is available at https://arxiv.org/abs/1711.03359.

Publisher Copyright:

© 2017 Keren Censor-Hillel and Michal Dory.

## Keywords

- Approximation algorithms
- Connectivity augmentation
- Distributed network design

## ASJC Scopus subject areas

- Software