The p-Center Problem with Connectivity Constraint

Let G(V, E, W) be a graph with n-vertex-set V and m-edge-set E in which each edge e is associated with a positive distance W(e). The p-Center problem is to locate some kind of facilities at p vertices of G to minimize the maximum distance between any vertex and the nearest facility corresponding to that vertex. This paper proposes an additional practical constraint. We restrict that the p vertices where the facilities are located must be connected, i.e., the subgraph induced by the p facility vertices must be connected. The resulting problem is called the Connected p-Center problem (the CpC problem). We first show that the CpC problem is NP-Hard on bipartite graphs and split graphs. Then, an O(n)-time algorithm for the problem on trees is proposed. Finally, we extend this algorithm to trees with forbidden vertices, i.e. some vertices in V cannot be selected as center vertices, and the time-complexity is also O(n).