The connected p-median problem on complete multi-layered graphs

We address in this paper the connected p-median problem on complete multi-layered graphs. We first solve the corresponding 1-median problem on the underlying graph in linear time based on the structure of the induced path. For p >= 2, we prove that the connected p-median contains at least one vertex in the layered set corresponding to the 1-median or its adjacent vertices in the induced path. Then, we develop an O(n log n) algorithm that solves the problem on a complete multi-layered graph with n vertices.