## Abstract

We consider network design problems in which we are given a graph G= (V, E) with edge costs, and seek a min-cost/size 2-node-connected subgraph G^{′}= (V^{′}, E^{′}) that satisfies a prescribed property. In the Block-Tree Augmentation problem the goal is to augment a tree T by a min-size edge set F⊆ E such that G^{′}= T∪ F is 2-node-connected. We break the natural ratio of 2 for this problem and show that it admits approximation ratio 1.91. This result extends to the related Crossing Family Augmentation problem.In the 2-Connected Dominating Set problem G^{′} should dominate V. We give the first non-trivial approximation algorithm for this problem, with expected ratio O~ (log^{4}| V| ).In the 2-Connected Quota Subgraph problem we are given node profits p(v) and G^{′} should have profit at least a given quota Q. We show expected ratio O~ (log^{2}| V| ), almost matching the best known ratio O(log^{2}| V| ). Our algorithms are very simple, and they combine three main ingredients: 1.A probabilistic spanning tree embedding with distortion O~ (log | V| ) results in a variant of the Block-Tree Augmentation problem.2.An approximation ratio preserving reduction of Block-Tree Augmentation variants to Node Weighted Steiner Tree problems.3.Using existing approximation algorithms for variants of the Node Weighted Steiner Tree problem.

Original language | English |
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Title of host publication | Approximation and Online Algorithms - 18th International Workshop, WAOA 2020, Revised Selected Papers |

Editors | Christos Kaklamanis, Asaf Levin |

Publisher | Springer Science and Business Media Deutschland GmbH |

Pages | 220-235 |

Number of pages | 16 |

ISBN (Print) | 9783030808785 |

DOIs | |

State | Published - 2021 |

Event | 18th International Workshop on Approximation and Online Algorithms, WAOA 2019 - Virtual, Online Duration: 9 Sep 2020 → 10 Sep 2020 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 12806 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 18th International Workshop on Approximation and Online Algorithms, WAOA 2019 |
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City | Virtual, Online |

Period | 9/09/20 → 10/09/20 |

### Bibliographical note

Publisher Copyright:© 2021, Springer Nature Switzerland AG.

## Keywords

- 2-connectivity
- Approximation algorithm
- Block-tree augmentation
- Dominating set
- Symmetric crossing family