Complexity of the critical node problem over trees

Abstract

In this paper we deal with the Critical Node Problem (CNP), i.e., the problem of searching for a given number K of nodes in a graph G, whose removal minimizes the (weighted or unweighted) number of connections between pairs of nodes in the residual graph. In particular, we study the case where the physical network represented by graph G has a hierarchical organization, so that G is a tree. The N P-completeness of this problem for general graphs has been already established (Arulselvan et al.). We study the subclass of CNP over trees, generalizing the objective function and constraints to take into account general nonnegative "costs" of node connections and "weights" for the nodes that are to be removed. We prove that CNP over trees is still N Pcomplete when general connection costs are specified, while the cases where all connections have unit cost are solvable in polynomial time by dynamic programming approaches. For the case with nonnegative connection costs and unit node weights we propose an enumeration scheme whose time complexity is within a polynomial factor from O(1.618034 n ), where n is the number of nodes of the tree. Results from computational experiments are reported for all the proposed algorithms.

Identifying critical nodes in undirected graphs: Complexity results and polynomial algorithms for the case of bounded treewidth

“…The algorithm runs in polynomial time for the class of graphs with treewidth bounded by a given constant, which include, among the others, the trees, all series-parallel graphs, all outerplanar graphs and Halin graphs (a longer list is given in [6]). This generalizes and extends the results given in [12] for the case of a tree. Finally, we show in Section 4 that the same dynamic programming scheme can be adapted to handle the different objective functions mentioned above, as well as certain edge-deletion (instead of node-deletion) problems.…”

“…Also, we need that the value of Ψ(α ′ , m ′ ) can be computed in polynomial time. (This function plays the role of that defined by equation (12). )…”

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, &.…”

Heuristic identification of critical nodes in sparse real-world graphs

“…Relatively few exact and heuristic methods have been proposed for the critical node detection problem (Borgatti 2006;Dinh et al 2010Dinh et al , 2012Summa et al 2011Summa et al , 2012Shen et al 2013;Addis et al 2013) and most have addressed small or special graphs (e.g. trees).…”