Optimizing the maximum vertex coverage attacks under knapsack constraint

Only when we understand how hackers think, can we defend against their attacks. Towards this end, this paper studies the cyber-attacks that aim to remove nodes or links from network topologies. We particularly focus on one type of such attacks called Maximum Vertex Coverage Attacks under Knapsack constraint (MVCAK), in which a hacker has a fixed budget to remove nodes from a network with the nodes involving different costs for removal, and the hacker's goal is to maximize the number of links incident to the nodes removed. Since the MVCAK problem is NP-hard, we firstly propose an optimal solution by Integer Linear Program formulation. Secondly, we give an approximate solution by Linear Programming relaxation that achieves an approximation ratio of 3/4, outperforming the existing 1 - 1/sqrt(e) (about 0.39). Thirdly, since the straightforward implementation of our approximate solution has a high time complexity, we propose two heuristics to significantly reduce its complexity while preserving the approximation ratio. We formally prove the correctness and the effectiveness of these two heuristics. Finally, we conduct extensive experiments on both artificial and real-world networks, showing that our approximate solution produces almost the same results as the optimal solution in practice and has an acceptable running time.