### Abstract

Given an undirected graph on n vertices with weights on its edges, Min WCF(p) consists of computing a covering forest of minimum weight such that each of its tree components contains at least p vertices. It has been proved that Min WCF(p) is NP-hard for any p≥4 (Imielinska et al., 1993) but -approximable (Goemans and Williamson, 1995). While Min WCF(2) is polynomial-time solvable, already the unweighted version of Min WCF(3) is NP-hard even on planar bipartite graphs of maximum degree 3. We prove here that for any p≥4, the unweighted version is NP-hard, even for planar bipartite graphs of maximum degree 3; moreover, the unweighted version for any p≥3 has no ptas for bipartite graphs of maximum degree 3. The latter theorem is the first-ever APX-hardness result on this problem. On the other hand, we show that Min WCF(p) is polynomial-time solvable on graphs with bounded treewidth, and for any p bounded by it has a ptas on planar graphs.

Original language | English |
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Title of host publication | Algorithms and Computation - 20th International Symposium, ISAAC 2009, Proceedings |

Publisher | Springer Verlag |

Pages | 892-901 |

Number of pages | 10 |

ISBN (Print) | 3642106307, 9783642106309 |

DOIs | |

Publication status | Published - Jan 1 2009 |

Event | 20th International Symposium on Algorithms and Computation, ISAAC 2009 - Honolulu, HI, United States Duration: Dec 16 2009 → Dec 18 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 | 5878 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | 20th International Symposium on Algorithms and Computation, ISAAC 2009 |
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Country | United States |

City | Honolulu, HI |

Period | 12/16/09 → 12/18/09 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Algorithms and Computation - 20th International Symposium, ISAAC 2009, Proceedings*(pp. 892-901). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 5878 LNCS). Springer Verlag. https://doi.org/10.1007/978-3-642-10631-6_90