  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
the 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 IC and LT 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 $(1-1/e-\epsilon)$-approximation in the problem under the LT model.
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.
