When attacking the network, people hope to get the maximum damage effect under the limited resources, and the node sorting strategy cannot achieve the maximum damage. In view of this situation, the network damage maximization problem of attacking a finite node set is defined, and an approximate algorithm for solving the problem is given. Due to the high computational complexity of the approximate algorithm, a greedy algorithm based on important nodes(GABIN) is proposed. Experiments on scale-free networks show that the GABIN algorithm can effectively reduce computing time, and its effect is close to the approximate algorithm. When the degree exponent of scale-free networks is greater than or equal to 2.5, the GABIN algorithm is better than sorting algorithms, and more than 30% of the nodes in the node set are different from sorting algorithms. The damage experiments on the Power grid show that the GABIN algorithm is suitable for large-scale real networks, and its effect is significantly better than sorting algorithms such as degree, betweenness, closeness and deleting nodes. The experiment shows that the key node set obtained by using the GABIN algorithm contains a large number of non-central nodes, which provides a new idea for network attack or network protection.