A simple greedy approximation algorithm for the minimum connected k -Center problem

In this paper, we consider the connected k -Center ( CkC ) problem, which can be seen as the classic k -Center problem with the constraint of internal connectedness, i.e., two nodes in a cluster are required to be connected by an internal path in the same cluster. CkC was first introduced by Ge et al. (ACM Trans Knowl Discov Data 2:7, 2008 ), in which they showed the NP -completeness for this problem and claimed a polynomial time approximation algorithm for it. However, the running time of their algorithm might not be polynomial, as one key step of their algorithm involves the computation of an NP -hard problem. We first present a simple polynomial time greedy-based 2 -approximation algorithm for the relaxation of CkC —the CkC^* . Further, we give a 6 -approximation algorithm for CkC .