Given a network with an ongoing epidemic, the network immunization problem seeks to identify a fixed number of nodes to immunize in order to maximize the number of infections prevented. A fundamental computational challenge in network immunization is that the objective function is generally neither submodular nor supermodular. Consequently, no efficient algorithm is known to consistently achieve a constant-factor approximation. Traditionally, this problem is partially addressed using proxy objectives that offer better approximation properties, but these indirect optimizations often introduce losses in effectiveness due to gaps between the proxy and natural objectives. In this paper, we overcome these fundamental barriers by leveraging the underlying stochastic structure of the diffusion process. Similar to the traditional influence objective, the immunization objective is an expectation expressed as a sum over deterministic instances. However, unlike the former, some of these terms are not submodular. Our key step is to prove that this sum has a bounded deviation from submodularity, enabling the classic greedy algorithm to achieve a constant-factor approximation for any sparse cascading network. We demonstrate that this approximation holds across various immunization settings and spread models.

翻译：给定一个存在流行病的网络，网络免疫问题旨在识别固定数量的节点进行免疫，以最大化预防的感染数量。网络免疫中的一个基本计算挑战在于目标函数通常既非子模也非超模。因此，目前尚未发现任何高效算法能够持续实现常数因子近似。传统上，这一问题通常通过采用具有更好近似性质的代理目标来部分解决，但这些间接优化方法往往因代理目标与自然目标之间的差距而导致有效性损失。本文通过利用扩散过程的底层随机结构，克服了这些根本性障碍。与传统的影响力目标类似，免疫目标是一个表示为确定性实例之和的期望。然而，与前者不同，其中某些项并不具有子模性。我们的关键步骤是证明该和与子模性的偏离有界，从而使经典贪心算法能够在任何稀疏级联网络上实现常数因子近似。我们证明了该近似结果适用于多种免疫设置和传播模型。