### Abstract

Let H be a laminar family of subsets of a groundset V.A k-cover of H is a multiset C of edges on V such that for every subset S in H, C has at least k edges that have exactly one end in S.A k-packing of H is a multiset P of edges on V such that for every subset S in H, P has at most k • u(S) edges that have exactly one end in S. Here, u assigns an integer capacity to each subset in H. Our main results are: (a) Given a k-cover C of H, there is an efficient algorithm to find a 1-cover contained in C of size <k\C\/(2k — 1). For 2-covers, the factor of 2/3 is best possible. (b) Given a 2-packing P of H, there is an efficient algorithm to find a 1-packing contained in P of size ≥|P|/3. The factor of 1/3 for 2-packings is best possible. These results are based on efficient algorithms for finding appropriate colorings of the edges in a k-cover or a 2-packing, respectively, and they extend to the case where the edges have nonnegative weights. Our results imply approximation algorithms for some NP-hard problems in connectivity augmentation and related topics. In particular, we have a 4/3-approximation algorithm for the following problem: Given a tree T and a set of nontree edges E that forms a cycle on the leaves of T, find a minimum-size subset E' of E such that T + E' is 2-edge connected.

Original language | English |
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Title of host publication | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |

Publisher | Springer Verlag |

Pages | 510-520 |

Number of pages | 11 |

Volume | 1643 |

ISBN (Print) | 3540662510, 9783540662518 |

DOIs | |

Publication status | Published - 1999 |

Event | 7th Annual European Symposium on Algorithms, ESA 1999 - Prague, Czech Republic Duration: Jul 16 1999 → Jul 18 1999 |

### 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 | 1643 |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 7th Annual European Symposium on Algorithms, ESA 1999 |
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Country | Czech Republic |

City | Prague |

Period | 7/16/99 → 7/18/99 |

### ASJC Scopus subject areas

- Computer Science(all)
- Theoretical Computer Science

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## Cite this

*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)*(Vol. 1643, pp. 510-520). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1643). Springer Verlag. https://doi.org/10.1007/3-540-48481-7_44