The obnoxious center problem on weighted cactus graphs

Zmazek, Blaž; Žerovnik, Janez

doi:10.1016/s0166-218x(03)00452-9

Cited by 29 publications

(17 citation statements)

References 14 publications

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“…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.…”

Section: Key Ideas Carried Forward: Center Problemsmentioning

confidence: 94%

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Hub Location Problems: The Location of Interacting Facilities

Kara

Taner

2011

International Series in Operations Research &Amp; Management Science

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), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. PrefaceThis book is the final result of a number of incidents that occurred to us while discussing location issues with colleagues at conferences. Frequently we attended presentations, in which the authors quoted the well-known references that helped to make the discipline what it is today. Upon further inquiry, though, it turned out that some of these colleagues had never actually read the original papers. We then discussed among ourselves which contributions could be credited with shaping the field. And, lo and behold, we found that we, too, had neglected to read some of the papers that form the foundation of our science. Whether it was laziness or other things that got in the way, it had become clear that something had to be done. Our first thought was to collect the original contributions (once we could agree on what they were) and reprint them. When discussing this possibility with a publisher, we immediately ran into a roadblock in the form of copyright. While this appeared to have stopped our enthusiastic effort dead in its tracks, we kept on collecting and reading what we considered original contributions.This went on until we met Camille Price, who suggested that, rather than reprinting the original contributions, we should invite some of the leaders in our field and ask them to describe the original contribution, explain and interpret it, and comment on the impact that it had to the field. This was, of course, an excellent idea, and the response by our colleagues to our pertinent requests was equally enthusiastic. What you hold in your hands is the result of this effort.In other words, the purpose of this book is to provide easy access to the main contributions to location theory. The book is organized as follows. The introductory chapter provides an overview of some of the many facets of location analysis. This is followed by contributions in the main three fields of inquiry: minisum, minimax, and covering problems. The next chapters are part of an ever-growing list of nonstandard location models: models including competitive components, those that locate undesirable facilities, those with probabilistic features, and those that allow interactions between facilities. The following chapters discuss solution techniques: after a discussion of exact and heuristic techniques, we devote an entire chapter to Weiszfeld's method, and another to Lagrangean techniques. The last chapters of this book deal with the spheres of influence that the facilities generate and that attract viii customers to them, an...

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Section: Key Ideas Carried Forward: Center Problemsmentioning

confidence: 94%

“…Zmazek and Zerovnik (2004) pose the E/V/1/TL weighted obnoxious center problem on cactus graphs. An O( c|V|) algorithm, where c denotes the number of different vertex weights, is given.…”

Section: Key Ideas Carried Forward: Center Problemsmentioning

confidence: 99%

Hub Location Problems: The Location of Interacting Facilities

Kara

Taner

2011

International Series in Operations Research &Amp; Management Science

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“…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.…”

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confidence: 99%

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confidence: 99%

“…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 Ben-Moshe, 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.…”

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Obnoxious Centers in Graphs

Cabello¹,

Rote²

2010

SIAM J. Discrete Math.

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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:

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