## Abstract

In the TREE AUGMENTATION problem the goal is to augment a tree T by a minimum size edge set F from a given edge set E such that T∪F is 2-edge-connected. The best approximation ratio known for the problem is 1.5. In the more general WEIGHTED TREE AUGMENTATION problem, F should be of minimum weight. WEIGHTED TREE AUGMENTATION admits several 2-approximation algorithms w.r.t. the standard cut-LP relaxation. Improving this natural ratio is a major open problem, and resolving it may have implications on other network design problems. It seems that achieving this goal requires finding an LP-relaxation with integrality gap better than 2, which is an old open problem even for TREE AUGMENTATION. In this paper we introduce two different LP-relaxations, and for each of them give a simple combinatorial algorithm that computes a feasible solution for TREE AUGMENTATION of size at most 1.75 times the optimal LP value.

Original language | English |
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Pages (from-to) | 94-105 |

Number of pages | 12 |

Journal | Discrete Applied Mathematics |

Volume | 239 |

DOIs | |

State | Published - 20 Apr 2018 |

### Bibliographical note

Funding Information:The first author’s research was supported in part by NSF grants 1218620 and 1540547 .

Publisher Copyright:

© 2018 Elsevier B.V.

## Keywords

- Approximation algorithm
- LP-relaxation
- Laminar family
- Tree augmentation