A ‘maximum node clustering’ problem

In this note we introduce a graph problem, called Maximum Node Clustering (MNC). We prove that the problem (which is easily shown to be strongly NP-complete) can be approximated in polynomial time within a ratio arbitrarily close to 2. For the special case where the graph is a tree, the problem is NP-complete in the ordinary sense; for this case we present a pseudopolynomial algorithm based on dynamic programming, and a related Fully Polynomial Time Approximation Scheme (FPTAS). Also, the tree case is shown to be exactly solvable in O(2^(2/3)npoly(n)) time, where n is the number of nodes.