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) [10], ) but -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 it has a ptas on planar graphs.