Towards proactive rumor control: When a budget constraint meets impression counts

The proliferation of rumors in online networks poses significant public safety risks and economic repercussions. Addressing this, we investigate the understudied aspect of rumor control: the interplay between influence block effect and user impression counts under budget constraints. We introduce two problem variants, RCIC and RCICB, tailored for diverse application contexts. Our study confronts two inherent challenges: the NP-hard nature of the problems and the non-submodularity of the influence block, which precludes direct greedy approaches. We develop a novel branch-and-bound framework for RCIC, achieving a ($1-1/e-ϵ$) approximation ratio, and enhance its efficacy with a progressive upper bound estimation, refining the ratio to ($1-1/e-ϵ-\rho$). Extending these techniques to RCICB, we attain approximation ratios of ($\frac{1}{2}(1-1/e)-ϵ$) and ($\frac{1}{2}(1-1/e-\rho )-ϵ$). We conduct experiments on real-world datasets to verify the efficiency, effectiveness, and scalability of our methods.