Parameterized complexity of critical node cuts

Abstract

We consider the following graph cut problem called Critical Node Cut (CNC): Given a graph G on n vertices, and two positive integers k and x, determine whether G has a set of k vertices whose removal leaves G with at most x connected pairs of vertices. We analyze this problem in the framework of parameterized complexity. That is, we are interested in whether or not this problem is solvable in f (κ)·n O(1) time (i.e., whether or not it is fixed-parameter tractable), for various natural parameters κ. We consider four such parameters:• The size k of the required cut.• The upper bound x on the number of remaining connected pairs.• The lower bound y on the number of connected pairs to be removed.• The treewidth w of G.We determine whether or not CNC is fixed-parameter tractable for each of these parameters. We determine this also for all possible aggregations of these four parameters, apart from w + k. Moreover, we also determine whether or not CNC admits a polynomial kernel for all these parameterizations. That is, whether or not there is an algorithm that reduces each instance of CNC in polynomial time to an equivalent instance of size κ O(1) , where κ is the given parameter. arXiv:1503.06321v2 [cs.DS] 28 Jun 2015vaccinating an area corresponds to deleting a certain vertex from the graph. Thus, the objective of stopping the virus from spreading translates to minimizing the number of connected pairs (two vertices which are in the same component) in the corresponding graph after applying the vaccination.This scenario can be modeled by the following graph-theoretic problem which we call Critical Node Cut (CNC). In this problem, we are given an undirected simple graph G and two integers k and x. The objective is to determine whether there exists a set C ⊆ V (G) of at most k vertices in G, such that the graph G − C which results from removing C from G, contains at most x connected pairs. In this sense, the cut C is considered critical since removing it from G leaves few (at most x) connected pairs. For convenience purposes, throughout the paper we will count ordered connected pairs; i.e., pairsThe goal of CNC is thus, roughly speaking, to destroy the connectivity of a given graph as much as possible given a certain budget for deleting vertices. From this point of view, CNC fits nicely to the broad family of graph-cut problems. Graph-cut problems have been studied widely and are among the most fundamental problems in algorithmic research. Examples include Min Cut, Max Cut, Multicut, Multiway Cut, Feedback Vertex Set, and Vertex Cover (see e.g.[21] for definitions of these problems). The latter is the special case of CNC with x = 0. Since Vertex Cover is arguably the most important problem in the theory of algorithmic design for NP-hard problems, CNC provides a natural test bed to see which of the techniques from this theory can be extended, and to what extent.Previous Work and Applications. The CNC problem has been studied from various different angles. The problem was shown to be NP-complete in [3] (although its NP-compl...