Complexity and approximation of the Constrained Forest problem

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− 1 n )-approximable (Goemans andWilliamson, 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( log n log log n ) it has a ptas on planar graphs.