The Critical Node Problem (CNP) is concerned with identifying the critical nodes in a complex network. These nodes play a significant role in maintaining the connectivity of the network, and removing them can negatively impact network performance. CNP has been studied extensively due to its numerous real-world applications. Among the different versions of CNP, CNP-1a has gained the most popularity. The primary objective of CNP-1a is to minimize the pair-wise connectivity in the remaining network after deleting a limited number of nodes from a network. Due to the NP-hard nature of CNP-1a, many heuristic/metaheuristic algorithms have been proposed to solve this problem. However, most existing algorithms start with a random initialization, leading to a high cost of obtaining an optimal solution. To improve the efficiency of solving CNP-1a, a knowledge-guided genetic algorithm named K2GA has been proposed. Unlike the standard genetic algorithm framework, K2GA has two main components: a pretrained neural network to obtain prior knowledge on possible critical nodes, and a hybrid genetic algorithm with local search for finding an optimal set of critical nodes based on the knowledge given by the trained neural network. The local search process utilizes a cut node-based greedy strategy. The effectiveness of the proposed knowledgeguided genetic algorithm is verified by experiments on 26 realworld instances of complex networks. Experimental results show that K2GA outperforms the state-of-the-art algorithms regarding the best, median, and average objective values, and improves the best upper bounds on the best objective values for eight realworld instances.

翻译：关键节点问题（CNP）旨在识别复杂网络中的关键节点。这些节点对维持网络连通性具有重要作用，删除它们会严重影响网络性能。由于其在现实世界中的广泛应用，CNP已得到广泛研究。在CNP的不同变体中，CNP-1a最受关注。CNP-1a的主要目标是从网络中删除有限节点后，最小化剩余网络中的成对连通性。鉴于CNP-1a的NP-hard性质，许多启发式/元启发式算法被提出以解决该问题。然而，现有算法大多从随机初始化开始，导致获取最优解的成本较高。为提高求解CNP-1a的效率，本文提出了一种知识引导的遗传算法K2GA。与标准遗传算法框架不同，K2GA包含两个主要部分：用于获取潜在关键节点先验知识的预训练神经网络，以及基于训练神经网络提供的知识，结合局部搜索寻找最优关键节点集的混合遗传算法。局部搜索过程采用基于割节点的贪心策略。通过在26个真实复杂网络实例上的实验，验证了所提知识引导遗传算法的有效性。实验结果表明，K2GA在最优值、中位数和平均值方面均优于现有最优算法，并改进了八个真实世界实例中最佳目标值的最优上界。