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

TL;DR

This paper investigates the limits of approximating influence maximization in social networks, showing that certain network structures and influence functions make the problem hard to approximate, but also providing exact algorithms for specific cases.

Contribution

It introduces strong inapproximability results for hierarchical blockmodel networks and 2-quasi-submodular influence functions, and offers a polynomial-time algorithm for a directed influence maximization variant.

Findings

Inapproximability results for hierarchical blockmodel networks.

Inapproximability for 2-quasi-submodular influence functions.

A polynomial-time algorithm for directed influence maximization on hierarchical blockmodels.

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 $N^{1-\varepsilon}$. This paper studies whether this worst-case hardness result can be circumvented by making assumptions about either the underlying network topology or the cascade model. All of 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-program 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.