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Covering a Graph with a Constrained Forest

Bazgan, Cristina; Couëtoux, Basile; Tuza, Zsolt (2009), Covering a Graph with a Constrained Forest, in Yingfei Dong, Ding-Zhu Du, Oscar Ibarra, Algorithms and Computation 20th International Symposium, ISAAC 2009, Honolulu, Hawaii, USA, December 16-18, 2009. Proceedings, Springer : Berlin Heidelberg, p. 892-901. 10.1007/978-3-642-10631-6_90

Type
Communication / Conférence
Date
2009
Conference country
UNITED STATES
Book title
Algorithms and Computation 20th International Symposium, ISAAC 2009, Honolulu, Hawaii, USA, December 16-18, 2009. Proceedings
Book author
Yingfei Dong, Ding-Zhu Du, Oscar Ibarra
Publisher
Springer
Published in
Berlin Heidelberg
ISBN
978-3-642-10630-9
Pages
892-901
Publication identifier
10.1007/978-3-642-10631-6_90
Metadata
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Author(s)
Bazgan, Cristina
Couëtoux, Basile
Tuza, Zsolt
Abstract (EN)
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 $(2-\frac{1}{n})$-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 $O(\frac{\log n}{\log\log n})$ it has a ptas on planar graphs.
Subjects / Keywords
graphs

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