We study the influence minimization problem: given a graph G and a seed set S, blocking at most b nodes or b edges such that the influence spread of seed set is minimized. This is a pivotal yet underexplored aspect of network analytics, which can limit the spread of undesirable phenomena in networks, such as misinformation and epidemics. Given the inherent NP-hardness of the problem under the independent cascade and linear threshold models, previous studies have employed greedy algorithms and Monte Carlo Simulations for its resolution. However, existing techniques become cost-prohibitive when applied to large networks due to the necessity of enumerating all the candidate blockers and computing the decrease in expected spread from blocking each of them. This significantly restricts the practicality and effectiveness of existing methods, especially when prompt decision making is crucial. In this paper, we propose the AdvancedGreedy algorithm, which utilizes a novel graph sampling technique that incorporates the dominator tree structure. We find that AdvancedGreedy can achieve a [Formula: see text]-approximation in the problem under the linear threshold model. For the problem under the independent cascade model, we further propose a novel heuristic algorithm GreedyReplace, based on identifying the relationships among candidate blockers. Experimental evaluations on real-life networks reveal that our proposed algorithms exhibit a significant enhancement in efficiency, surpassing the state-of-the-art algorithm by three orders of magnitude while achieving high effectiveness. History: Accepted by Erwin Pesch, Area Editor for Heuristic Search & Approximation Algorithms. Funding: K. Wang was supported by the National Natural Science Foundation of China [Grants 72221001 and 62302294]. J. Liu was supported by the National Natural Science Foundation of China [Grants 72221001 and 72402127] and Shanghai Pujiang Program [Grant 23PJC059]. F. Zhang was supported by the Guangdong Basic and Applied Basic Research Foundation [Grants 2023A1515012603 and 2024A1515011501]. W. Zhang was supported by the Australian Research Council [Grant FT210100303]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0591 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0591 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .