# Complexity and approximation of the Constrained Forest problem

Bazgan, Cristina and Couëtoux, Basile and Tuza, Zsolt
(2011)
*Complexity and approximation of the Constrained Forest problem.*
THEORETICAL COMPUTER SCIENCE, 412 (32).
pp. 4081-4091.
ISSN 0304-3975

## 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. © 2010 Elsevier B.V. All rights reserved.

Item Type: | ISI Article |
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Uncontrolled Keywords: | Treewidth, Planar graphs, Bipartite graphs, PTAS, Constrained Forest, APX-hardness, APPROXIMATION, COMPLEXITY |

Subjects: | Q Science > QA Mathematics and Computer Science > QA75 Electronic computers. Computer science / számítástechnika, számítógéptudomány |

Depositing User: | EPrints Admin |

Date Deposited: | 05 Feb 2014 12:16 |

Last Modified: | 05 Feb 2014 15:47 |

URI: | http://eprints.sztaki.hu/id/eprint/7571 |

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