Complexity of Determining the Most Vital Elements for the 1-median and 1-center Location Problems

Abstract

Abstract. We consider the k most vital edges (nodes) and min edge (node) blocker versions of the 1-median and 1-center location problems. Given a weighted connected graph with distances on edges and weights on nodes, the k most vital edges (nodes) 1-median (respectively 1-center) problem consists of finding a subset of k edges (nodes) whose removal from the graph leads to an optimal solution for the 1-median (respectively 1-center) problem with the largest total weighted distance (respectively maximum weighted distance). The complementary problem, min edge (node) blocker 1-median (respectively 1-center), consists of removing a subset of edges (nodes) of minimum cardinality such that an optimal solution for the 1-median (respectively 1-center) problem has a total weighted distance (respectively a maximum weighted distance) at least as large as a specified threshold. We show that k most vital edges 1-median and k most vital edges 1-center are NP -hard to approximate within a factor − ǫ respectively, for any ǫ > 0, while k most vital nodes 1-median and k most vital nodes 1-center are NP -hard to approximate within a factor 3 2 − ǫ, for any ǫ > 0. We also show that the complementary versions of these four problems are NP -hard to approximate within a factor 1.36.

Detecting critical node structures on graphs: A mathematical programming approach

“…However, despite the broad interest on the subject, there seems to be no overall agreement on a base name that encompasses and describes the general case. Most authors often use the names such as critical elements problems [6], network interdiction problems [38], node (edge) deletion problems [55], most vital elements problems [10], key players identification [14], blocker problems [46], and disruptor problems [23,65].…”

“…In many of these studies, a wide variety of structural properties and their corresponding measures have been proposed to quantify the disruption inflicted on the graph G. In general, these measures are either associated with (1) optimal solutions to flow problems over G, such as shortest paths, maximum flow, or minimum cost flow problems [17,19,33,38,41,44,49,78,79]; (2) the sizes or relative weight of some topological node or edge structures in G, like spanning trees [28], dominating sets [47], central nodes-for example, the one-median or one-center nodes [10], matchings [80,81], independent sets [9], cliques [46], and node covers [9]; and (3) connectivity and cohesiveness properties of G, such as the total number of connected node pairs [1, 6, 21-23, 51, 52, 65, 69, 70, 72, 73], the weight of the connections between the node pairs [6,21,73], the size of the largest component [13,30,55,63,64,73], the total number of components [5,[63][64][65]68], distance-based connectivity metrics [74], and the graph information entropy [14,35,56,73].…”

Minimum edge blocker dominating set problem

“…The minimum vertex/edge blocker and the most vital vertices/ edges problems have been studied in literature with respect to different graph properties, such as connectivity (Addis, Di Summa, & Grosso, 2013;Arulselvan, Commander, Elefteriadou, & Pardalos, 2009;Di Summa, Grosso, & Locatelli, 2011;Shen, Smith, & Goli, 2012;Veremyev, Prokopyev, & Pasiliao, 2014), shortest path (Bar-Noy et al, 1995;Israeli & Wood, 2002;Khachiyan et al 2008), maximum flow (Altner, Ergun, & Uhan, 2010;Ghare, Montgomery, & Turner, 1971;Wollmer, 1964;Wood, 1993), spanning tree (Bazgan, Toubaline, & Vanderpooten, 2012;2013;Frederickson & Solis-Oba, 1996), assignment (Bazgan, Toubaline, & Vanderpooten, 2010b), 1-median (Bazgan et al, 2010a), 1-center (Bazgan et al, 2010a), matching (Ries et al, 2010;Zenklusen, 2010;Zenklusen et al, 2009), independent sets (Bazgan et al, 2011), vertex covers (Bazgan et al, 2011), andcliques (Mahdavi Pajouh, Boginski, &.…”

“…The most vital edges assignment problem (and minimum edge assignment blocker problem) has been shown to be NPhard, and hard to approximate within a factor of 2 (and within a factor of 1.36, respectively) if P = NP (Bazgan et al, 2010b). Bazgan et al (2010a) investigated the complexity of the most vital edges (vertices) 1-median (1-center) problem and the minimum edge (vertex) 1-median (1-center) blocker problem and showed that these problems are NP-hard to approximate within a factor c, for some c > 1.…”

“…In the related literature, the first problem is referred to as minimum vertex/edge blocker problem (Bazgan, Toubaline, & Tuza, 2011;Ries, Bentz, Picouleau, de Werra, Costa, & Zenklusen, 2010;Zenklusen, Ries, Picouleau, de Werra, Costa, & Bentz, 2009) and the second one is called most vital vertices/edges problem (Bar-Noy, Khuller, & Schieber, 1995;Bazgan et al, 2011;Bazgan, Toubaline, & Vanderpooten, 2010a). In this paper, we introduce the problem of detecting a smallest subset of edges in a weighted undirected graph whose removal bounds the minimum weight of a dominating set in the remaining graph.…”