Locating Facilities on a Network to Minimize Their Average Service Radius

Abstract

Let G = (V,E) denote an undirected weighted graph of n nodes and m edges, and let U ⊆ V. The relative eccentricity of a node v ∈ U is the maximum distance in G between v and any other node of U, while the radius of U in G is the minimum relative eccentricity of all the nodes in U. Several facility location problems ask for partitioning the nodes of G so as to minimize some global optimization function of the radii of the subsets of the partition. Here, we focus on the problem of partitioning the nodes of G into exactly p ≥ 2 non-empty subsets, so as to minimize the sum of the subset radii, called the total radius of the partition. This problem can be easily seen to be NP-hard when p is part of the input, but when p is fixed it can be solved in polynomial time by reducing it to a similar partitioning problem. In this paper, we first present an efficient O(n ³) time algorithm for the notable case p = 2, which improves the O(mn ² + n ³ logn) running time obtainable by applying the aforementioned reduction. Then, in an effort of characterizing meaningful polynomial-time solvable instances of the problem when p is part of the input, we show that (i) when G is a tree, then the problem can be solved in O(n ³ p ³) time, and (ii) when G has bounded treewidth h, then the problem can be solved in O(n ^4h + 4 p ³) time.

Work partially supported by the Research Project GRID.IT, funded by the Italian Ministry of University and Research, by the European Union under COST 295 (DYNAMO), and by the Swiss BBW under grant no. C.05.0047. Part of this work has been developed while the first and the third authors were visiting ETH.