Obnoxious Centers in Graphs

Abstract

Abstract. We consider the problem of finding obnoxious centers in graphs. For arbitrary graphs with n vertices and m edges, we give a randomized algorithm with O(n log 2 n + m log n) expected time. For planar graphs, we give algorithms with O(n log n) expected time and O(n log 3 n) worstcase time. For graphs with bounded treewidth, we give an algorithm taking O(n log n) worst-case time. The algorithms make use of parametric search and several results for computing distances on graphs of bounded treewidth and planar graphs.Key words. graph algorithms, facility location, planar graphs, parametric search, bounded treewidth AMS subject classifications. 05C85, 68W05, 90B851. Introduction. A central problem in locational analysis deals with the placement of new facilities that optimize a given objective function. In the obnoxious center problem, there is set of sites in some metric space, each with its own weight, and we want to place a facility that maximizes the minimum of the weighted distances from the given sites. The problem arises naturally when considering the placement of an undesirable facility that will affect the environment, or, in a dual setting, when searching for a place away from existing obnoxious facilities. Algorithmically, obnoxious facilities have received much attention previously; see [1,2,7,11,15,23,25,26,27] and references therein.In this paper, we consider the problem of placing a single obnoxious facility in a graph, either at its vertices or along its edges; this is often referred to as the continuous problem, as opposed to the discrete version, where the facility has to be placed in a vertex of G. A formal definition of the problem is given in Section 2.1. We use n, m for the number of vertices and edges of G, respectively.Previous results. Subquadratic algorithms are known for the obnoxious center problem in trees and cacti. Tamir [25] gave an algorithm with O(n log 2 n) worst-case time for trees. Faster algorithms are known for some special classes of trees [7,25]. For cactus graphs, Zmazek andŽerovnik [27] gave an algorithm using O(cn) time, where c is the number of different weights in the sites, and recently Bhattacharya, and Shi [1] showed an algorithm using O(n log 3 n) time. For general graphs, Tamir [24] showed how to solve the obnoxious center problem in O(nm + n 2 log n) time. We are not aware of other works for special classes of graphs. However, for planar graphs, it is easy to use separators of size O( √ n) [17] to solve the problem in roughly O(n 3/2 ) time. Our results. In general, we follow an approach similar to Tamir [25], using the close connection between the obnoxious center problem and the following covering problem: do a set of disks cover a graph? See Section 2.2 for a formal definition. A summary of our results is as follows:

Hub Location Problems: The Location of Interacting Facilities

“…Additional results for E/V/1/TL when the center objective is replaced by a weighted obnoxious center objective, are found in Zmazek and Zerovnik (2004) and Cabello and Rote (2007). Zmazek and Zerovnik (2004) pose the E/V/1/TL weighted obnoxious center problem on cactus graphs.…”

“…For the unweighted version c can be dropped and a linear time algorithm results. Cabello and Rote (2007) give an O( n log 3 n) algorithm that finds a weighted E/V/1/G obnoxious center on any planar graph. The complexity simplifies to O( n log n) for graphs with a bounded treewidth.…”