## Abstract

A graph G is k-out-connected from its node s if it contains k internally disjoint sv-paths to every node v; G is k-connected if it is k-out-connected from every node. In connectivity augmentation problems, the goal is to augment a graph G0 = (V, E0) by a minimum costs edge set J such that G0 ∪ J has higher connectivity than G0. In the k-Out-Connectivity Augmentation (k-OCA) problem, G0 is (k − 1)-out-connected from s and G0 ∪ J should be k-out-connected from s; in the k-Connectivity Augmentation (k-CA) problem G0 is (k − 1)-connected and G0 ∪ J should be k-connected. The parameterized complexity status of these problems was open even for k = 3 and unit costs. We will show that k-OCA and 3-CA can be solved in time 9^{p} · n^{O(1)}, where p is the size of an optimal solution. Our paper is the first that shows fixed-parameter tractability of a k-node-connectivity augmentation problem with high values of k. We will also consider the (2, k)-Connectivity Augmentation ((2, k)-CA) problem where G0 is (k − 1)-edge-connected and G0 ∪ J should be both k-edge-connected and 2-connected. We will show that this problem can be solved in time 9^{p} · n^{O(1)}, and for unit costs approximated within 1.892.

Original language | English |
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Title of host publication | 32nd Annual European Symposium on Algorithms, ESA 2024 |

Editors | Timothy Chan, Johannes Fischer, John Iacono, Grzegorz Herman |

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

ISBN (Electronic) | 9783959773386 |

DOIs | |

State | Published - Sep 2024 |

Event | 32nd Annual European Symposium on Algorithms, ESA 2024 - London, United Kingdom Duration: 2 Sep 2024 → 4 Sep 2024 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 308 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 32nd Annual European Symposium on Algorithms, ESA 2024 |
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Country/Territory | United Kingdom |

City | London |

Period | 2/09/24 → 4/09/24 |

### Bibliographical note

Publisher Copyright:© Zeev Nutov; licensed under Creative Commons License CC-BY 4.0.

## Keywords

- fixed-parameter tractability
- node connectivity augmentation