Dynamically Exchanging Nodes of the Critical Node Problem

Abstract

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.