Dismantle a network efficiently during the entire process by a compound algorithm

The dismantling network problem only asks the minimal vertex set of a graph after removing which the remaining graph will break into connected components of sub-extensive size, but we should also consider the efficiency of intermediate states during the entire dismantling process, which is measured by the general performance R in this paper. In order to improve the general performance of the belief-propagation decimation (BPD) algorithm, we implement a compound algorithm (CA) mixing the BPD algorithm and the node explosive percolation (NEP) algorithm. In this CA, the NEP algorithm will rearrange and optimize the head part of a dismantling sequence of vertices which is given by the BPD algorithm firstly. Two ancestor algorithms are connected at the joint point where the general performance can be optimized. The numerical computations on ErdösRényi graphs, random-regular graphs, and scale-free graphs with various average degree prove the CA inherits the advantage of its two ancestors: It can dismantle a graph to small pieces as quickly as the BPD algorithm, and it is with the efficiency of the NEP algorithm during the entire dismantling process. By studying how the joint point of two ancestor algorithms affects the value of R, we find that a wise joint point is where the BPD algorithm breaks the original graph to subgraphs with the size just smaller than the 1% of the original one. We refer the CA with this settled joint point as the fast CA. The fast CA overcomes a drawback in the original CA that the original CA is in the higher complexity class than the NEP algorithm. The numerical computations on random graph ensembles with the size from 2 to 2 reveal that the fast CA is in the same complexity class with the BPD algorithm. The computation on some real-world instances also exhibits that using the fast CA to optimize the intermediate process of a dismantling algorithm is an effective approach indeed.