Clustered Trees with Minimum Inter-cluster Distance

For a given edge-weighted graph G = (V, E, w), in which the vertices are partitioned into clusters R = {R1, R2, ... , Rk}, a spanning tree of G is a clustered spanning tree if the subtrees spanning the clusters are mutually disjoint. In this paper we study the problem of constructing a clustered spanning tree such that the total distance summed over all vertices of different clusters is minimized. We show that the problem is polynomial-time solvable when the number of clusters k is 2 and NP-hard for k = 3. We also present a 2-approximation algorithm for the case of 3 clusters.