Beyond Worst-case (In)approximability of Nonsubmodular Influence Maximization

Abstract

We consider the problem of maximizing the spread of influence in a social network by choosing a fixed number of initial seeds, formally referred to as the influence maximization problem . It admits a (1−1/e)-factor approximation algorithm if the influence function is submodular . Otherwise, in the worst case, the problem is NP-hard to approximate to within a factor of N 1−ε . This article studies whether this worst-case hardness result can be circumvented by making assumptions about either the underlying network topology or the cascade model. All our assumptions are motivated by many real-life social network cascades. First, we present strong inapproximability results for a very restricted class of networks called the (stochastic) hierarchical blockmodel , a special case of the well-studied (stochastic) blockmodel in which relationships between blocks admit a tree structure. We also provide a dynamic-programming-based polynomial time algorithm, which optimally computes a directed variant of the influence maximization problem on hierarchical blockmodel networks. Our algorithm indicates that the inapproximability result is due to the bidirectionality of influence between agent-blocks. Second, we present strong inapproximability results for a class of influence functions that are “almost” submodular, called 2-quasi-submodular . Our inapproximability results hold even for any 2-quasi-submodular f fixed in advance. This result also indicates that the “threshold” between submodularity and nonsubmodularity is sharp, regarding the approximability of influence maximization.