Complexity and approximation of the Constrained Forest problem

Cristina Bazgan, Basile Couëtoux, Zsolt Tuza

Research output: Contribution to journalArticle

7 Citations (Scopus)


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 languageEnglish
Pages (from-to)4081-4091
Number of pages11
JournalTheoretical Computer Science
Issue number32
Publication statusPublished - Jul 22 2011


  • 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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