Algorithms for connected p-centdian problem on block graphs

Abstract

We consider the facility location problem of locating a set \(X_p\) of p facilities (resources) on a network (or a graph) such that the subnetwork (or subgraph) induced by the selected set \(X_p\) is connected. Two problems on a block graph G are proposed: one problem is to minimizes the sum of its weighted distances from all vertices of G to \(X_p\), another problem is to minimize the maximum distance from each vertex that is not in \(X_p\) to \(X_p\) and, at the same time, to minimize the sum of its distances from all vertices of G to \(X_p\). We prove that the first problem is linearly solvable on block graphs with unit edge length. For the second problem, it is shown that the set of Pareto-optimal solutions of the two criteria has cardinality not greater than n, and can be obtained in \(O(n^2)\) time, where n is the number of vertices of the block graph G.