Optimal Detection of Critical Nodes: Improvements to Model Structure and Performance

The graph partitioning problem (GPP) consists of partitioning the vertex set of a graph into several disjoint subsets so that the sum of weights of the edges between the disjoint subsets isminimized. The critical node problem (CNP) is to detect a set of vertices in a graph whose deletion results in the graph having the minimum pairwise connectivity between the remaining vertices. Both GPP and CNP find many applications in identification of community structures or influential individuals in social networks, telecommunication networks, and supply chain networks. In this paper, we use integer programming to formulate GPP and CNP. In several practice cases, we have networks with uncertain weights of links. Some times, these uncertainties have no information of probability distribution. We use robust optimization models of GPP and CNP to formulate the community structures or influential individuals in such networks.

Component-cardinality-constrained critical node problem in graphs

With increasing significance of Wireless Sensor Networks (WSNs) on real-time applications, cluster formation and critical node avoidance plays a key role in determining the performance of WSNs. The sensor node that cannot join as a cluster member with any cluster head is denoted as critical node. The effectiveness of clustering mechanism could be improved by minimizing the critical node occurrence and the link changes. Numerous single hop and distributed cluster formation algorithms have been proposed in the literature to develop clusters. However, the existing algorithms do not have proper solutions to handle link failures and critical node problem in mobile sensor environment. A novel protocol, the Velocity Energy-Efficient and Link Stability-based Clustering Protocol (VELSCP) has been proposed in this paper to effectively overcome the critical node problem and to form clusters with stable links. The proposed protocol successfully addresses the critical node problem and link failure issues by selecting a proper cluster head (CH). In this methodology, a sensor node (SN) selects itself as a cluster head among one hop neighbor nodes based on its residual energy, coverage distance, Received Signal Strength (RSS), mobility and connection time. The proposed scheme is evaluated and compared with established algorithms like LEACH, HEED and MBC. Simulation results clearly demonstrate that the proposed scheme provides better QoS than the existing schemes in terms of connection time, RSS, throughput, delay and energy consumption.

ADA-PC: An asynchronous distributed algorithm for minimizing pairwise connectivity in wireless multi-hop networks

We deal with the critical node problem (CNP) in a graph [Formula: see text], in which a given number [Formula: see text] of nodes are removed to minimize the connectivity of the residual graph in some sense. Several ways to minimize some connectivity measurement have been proposed, including minimizing the connectivity index(MinCI), maximizing the number of components, minimizing the maximal component size. We propose two classes of CNPs by combining the above measurements together. The objective is to minimize the sum of connectivity indexes and the total degrees in the residual graph. The CNP with an upper-bound [Formula: see text] on the maximal component size is denoted by MSCID-CS and the one with an extra upper-bound [Formula: see text] on the number of components is denoted by MSCID-CSN. They are generalizations of the MinCI, which has been shown NP-hard for general graphs. In particular, we study the case where [Formula: see text] is a tree. Two dynamic programming algorithms are proposed to solve the two classes of CNPs. The time complexities of the algorithms for MSCID-CS and MSCID-CSN are [Formula: see text] and [Formula: see text], respectively, where [Formula: see text] is the number of nodes in [Formula: see text]. Computational experiments are presented which show the effectiveness of the algorithms.

In this paper, we propose a polynomial-time algorithm for solving the Component-Cardinality-Constrained Critical Node Problem (3C-CNP) on bipartite permutation graphs. This problem, which is a variant of the well-known Critical Node Detection problem, consists in finding the minimal subset of nodes within a graph, the deletion of which results in a set of connected components of at most K nodes each one, where K is a given integer. The proposed algorithm is a dynamic programming scheme of time complexity $$O(nK^2)$$ , where n is the number of nodes. To provide evidences of algorithm’s efficiency, different experiments have been performed on randomly generated graphs.

Efficient heuristic algorithm for identifying critical nodes in planar networks

The critical node problem (CNP) aims to identify a subset of critical nodes in an undirected graph such that removing these critical nodes minimizes the pairwise node connectivity over the residual graph. CNP has various applications; however, it is computationally challenging. This paper introduces FastCNP, a fast heuristic algorithm for solving the problem. FastCNP employs an effective two-phase node exchange strategy to locate high-quality solutions and applies a destructive-constructive perturbation procedure to drive the search to new regions when the search stagnates. Computational results on 16 popular benchmark instances show that FastCNP finds improved best results (new upper bounds) for 6 instances, and matches the best-known results for 9 instances.

The Critical Node Problem (CNP) aims to identify k nodes from an undirected graph G=(V,E), in order to minimize the number of pairwise connected nodes in the residual graph after deleting these k nodes. This problem has a wide range of applications in the fields of cyber security, disease control, biological analysis, social network, etc. The CNP is known to be NP-hard, and heuristics are commonly used to solve large-scale CNP instances, among which the node-exchange operation is a basic move operator widely adopted by local-search based heuristics. Given an initial CNP solution consisting of k nodes, there are totally k×(|V|−k) possible node-exchange operations. The current best algorithm requires a complexity of O((|V|−k)×(|V|+|E|+k×D(G))) to evaluate all these operations (|V| denotes the number of nodes, |E| denotes the number of edges, D(G) denotes the maximum degree of the node in graph G).In this paper, a series of dynamic data structures are implemented to reduce the above complexity to O((|V|−k)×(|V|+|E|)), so as to improve the efficiency of local search.

The critical node detection problems (CNDPs) have important applications in network security, smart grid, epidemic control, drug design, and risk assessment. The critical node problem (CNP) is one of well-known CNDPs, which is NP-hard. In this work, we combine frequent pattern mining with evolutionary algorithm for solving CNP, where pattern mined from high-quality solutions are used to guide the construction of offspring solution. More specifically, based on the memetic algorithm (Memetic Algorithm for CNP, MACNP), frequent pattern mining was integrated into the memetic algorithm framework. The frequent patterns mined were used to construct the offspring solution, instead of crossover operator in MACNP. In the research, the nodes in frequent items set were directly fixed as part of the offspring solution. So the offspring solution inherited excellent properties from more parent solutions, thus forming the algorithm called Frequent Pattern Based Search for CNP (FPBS-CNP). This strategy improved the algorithm efficiency. Experiments were conducted on synthetic and real-world benchmark. We experimentally compare FPBS-CNP with MACNP. The experiment results show that FPBS-CNP performs well on small instances and has potential in solving large instances.

The Connected Critical Node Problem

We tackle a stochastic version of the critical node problem (CNP) where the goal is to minimize the pairwise connectivity of a graph by attacking a subset of its nodes. In the stochastic setting considered, the outcome of attacks on nodes is uncertain. In our work, we focus on trees and demonstrate that over trees the stochastic CNP actually generalizes to the stochastic critical element detection problem where the outcome of attacks on edges is also uncertain. We prove the NP‐completeness of the decision version of the problem when connection costs are one, while its deterministic counterpart was proved to be polynomial. We then derive a nonlinear model for the considered CNP version over trees and provide a corresponding linearization based on the concept of probability chains. Moreover, given the features of the derived linear model, we devise an exact Benders decomposition (BD) approach where we solve the slave subproblems analytically. A strength of our approach is that it does not rely on any statistical approximation such as the sample average approximation, which is commonly employed in stochastic optimization. We also introduce an approximation algorithm for the problem variant with unit connection costs and unit attack costs, and a specific integer linear model for the case where all the survival probabilities of the nodes in case of an attack are equal. Our methods are capable of solving relevant instances of the problem with hundreds of nodes within 1 hour of computational time. With this work, we aim to foster research on stochastic versions of the CNP, a problem tackled mainly in deterministic contexts so far. Interestingly, we also show a successful application of the concept of probability chains for problem linearizations significantly improved by decomposition methods such as the BD.

In this paper we deal with the critical node problem, where a given number of nodes has to be removed from an undirected graph in order to maximize the disconnections between the node pairs of the graph. We propose an integer linear programming model with a non-polynomial number of constraints but whose linear relaxation can be solved in polynomial time. We derive different valid inequalities and some theoretical results about them. We also propose an alternative model based on a quadratic reformulation of the problem. Finally, we perform many computational experiments and analyze the corresponding results.

The distance-based critical node problem involves identifying a subset of nodes in a network whose removal minimises a pre-defined distance-based connectivity measure. Having the classical critical node problem as a special case, the distance-based critical node problem is computationally challenging. In this article, we study the distance-based critical node problem from a heuristic algorithm perspective. We consider the distance-based connectivity objective whose goal is to minimise the number of node pairs connected by a path of length at most k, subject to budgetary constraints. We propose a centrality based heuristic which combines a backbone-based crossover procedure to generate good offspring solutions and a centrality-based neighbourhood search to improve the solution. Extensive computational experiments on real-world and synthetic graphs show the effectiveness of the developed heuristic in generating good solutions when compared to exact solution. Our empirical results also provide useful insights for future algorithm development.

Detecting critical nodes in sparse graphs