## Abstract

We consider the (undirected) Node Connectivity Augmentation (NCA) problem: given a graph J = (V,E _{J}) and connectivity requirements {r(u,v):u,v ∈ V}, find a minimum size set I of new edges (any edge is allowed) so that J + I contains r(u,v) internally disjoint uv-paths, for all u,v ∈ V. In the Rooted NCA there is s ∈ V so that r(u,v) > 0 implies u = s or v = s. For large values of k = max _{u,v ∈V} r(u,v), NCA is at least as hard to approximate as Label-Cover and thus it is unlikely to admit a polylogarithmic approximation. Rooted NCA is at least as hard to approximate as Hitting-Set. The previously best approximation ratios for the problem were O(k ln n) for NCA and O(ln n) for Rooted NCA. In [Approximating connectivity augmentation problems, SODA 2005] the author posed the following open question: Does there exist a function ρ(k) so that NCA admits a ρ(k)-approximation algorithm? In this paper we answer this question, by giving an approximation algorithm with ratios O(k ln ^{2} k) for NCA and O(ln ^{2} k) for Rooted NCA. This is the first approximation algorithm with ratio independent of n, and thus is a constant for any fixed k. Our algorithm is based on the following new structural result which is of independent interest. If is a set of node pairs in a graph J, then the maximum degree in the hypergraph formed by the inclusion minimal tight sets separating at least one pair in D is O(ℓ ^{2}), where ℓ is the maximum connectivity of a pair in D.

Original language | English |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization |

Subtitle of host publication | Algorithms and Techniques - 12th International Workshop, APPROX 2009 and 13th International Workshop, RANDOM 2009, Proceedings |

Pages | 286-297 |

Number of pages | 12 |

DOIs | |

State | Published - 2009 |

Event | 12th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2009 and 13th International Workshop on Randomization and Computation, RANDOM 2009 - Berkeley, CA, United States Duration: 21 Aug 2009 → 23 Aug 2009 |

### 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 | 5687 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 12th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2009 and 13th International Workshop on Randomization and Computation, RANDOM 2009 |
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Country/Territory | United States |

City | Berkeley, CA |

Period | 21/08/09 → 23/08/09 |