(r, P)-centroid Problems on Paths and Trees

An instance of the (r, p)-centroid problem is given by an edge and node weighted graph. Two competitors, the leader and the follower, are allowed to place p or r facilities, respectively, into the graph. Users at the nodes connect to the closest facility. A solution of the (r, p)-centroid problem is a leader placement such that the maximum total weight of the users connecting to any follower placement is as small as possible. We show that the absolute (r, p)-centroid problem is NP-hard even on a path which answers a long standing open question of the complexity of the problem on trees (Hakimi, 1990). Moreover, we provide polynomial time algorithms for the discrete (r, p)-centroid on paths and the (1, p)-centroid on trees, and complementary hardness results for more complex graph classes.