Partitioning the nodes of a graph to minimize the sum of subgraph radii

Abstract

Let G=(V,E) denote a weighted graph of n nodes and m edges, and let G|V′| denote the subgraph of G induced by a subset of nodes . The radius of G|V′| is the maximum length of a shortest path in G|V′| emanating from its center (i.e., a node of G|V′| of minimum eccentricity). In this paper, we focus on the problem of partitioning the nodes of G into exactly p non-empty subsets, so as to minimize the sum of the induced subgraph radii. We show that this problem - which is of significance in facility location applications - is NP-hard when p is part of the input, but for a fixed constant p > 2 it can be solved in O(n2p/p!) time. Moreover, for the notable case p=2, we present an efficient O(m n2+n3 logn) time algorithm.© 2006 Springer-Verlag Berlin/Heidelberg.