We consider the influence maximization problem (IMP) which asks for identifying a limited number of key individuals to spread influence in a social network such that the expected number of influenced individuals is maximized. The stochastic maximal covering location problem (SMCLP) formulation is a mixed integer programming formulation that effectively approximates the IMP by the Monte-Carlo sampling. For IMPs with a large-scale network or a large number of samplings, however, the SMCLP formulation cannot be efficiently solved by existing exact algorithms due to its large problem size. In this paper, we concentrate on deriving presolving methods to reduce the problem size and hence enhance the capability of employing exact algorithms in solving large-scale IMPs. In particular, we propose two effective presolving methods, called strongly connected nodes aggregation (SCNA) and isomorphic nodes aggregation (INA), respectively. The SCNA enables to build a new SMCLP formulation that is potentially much more compact than the existing one, and the INA further eliminates variables and constraints in the SMCLP formulation. A theoretical analysis on two special cases of the IMP is provided to demonstrate the strength of the SCNA and INA in reducing the problem size of the SMCLP formulation. Finally, we integrate the proposed presolving methods, SCNA and INA, into the Benders decomposition algorithm, which is recognized as one of the state-of-the-art exact algorithms for solving the IMP. Numerical results demonstrate that with the SCNA and INA, the Benders decomposition algorithm is much more effective in solving the IMP in terms of solution time.
S.-J. Chen, W.-K. Chen, Y.-H. Dai, J.-H. Yuan, and H.-S. Zhang. Efficient presolving methods for the influence maximization problem in social networks. 2021.