### 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) [10]) but (2-1n)-approximable (Goemans and Williamson (1995) [9]). 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 O(lognloglogn) it has a ptas on planar graphs.

Original language | English |
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Pages (from-to) | 4081-4091 |

Number of pages | 11 |

Journal | Theoretical Computer Science |

Volume | 412 |

Issue number | 32 |

DOIs | |

Publication status | Published - Jul 22 2011 |

### Keywords

- APX-hardness
- Approximation
- Bipartite graphs
- Complexity
- Constrained Forest
- Planar graphs
- Ptas
- Treewidth

### ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science(all)

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## Cite this

*Theoretical Computer Science*,

*412*(32), 4081-4091. https://doi.org/10.1016/j.tcs.2010.07.018