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. One of the fundamental computational challenges in network immunization is that the objective function is generally neither submodular nor supermodular. As a result, no efficient algorithm is known to consistently find a solution with a constant approximation guarantee. Traditionally, this problem is addressed using proxy objectives, which offer better approximation properties. However, converting to these indirect optimizations often introduces losses in effectiveness. In this paper, we overcome these fundamental barriers by utilizing the underlying stochastic structures of the diffusion process. Similar to the traditional influence objective, the immunization objective is an expectation that can be expressed as the sum of objectives over deterministic instances. However, unlike the former, some of these terms are not submodular. The key step is proving that this sum has a bounded deviation from submodularity, thereby enabling the greedy algorithm to achieve constant factor approximation. We show that this approximation still stands considering a variety of 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，目前尚未有已知的高效算法能够持续找到具有常数近似保证的解。传统上，该问题通过使用具有更好近似性质的代理目标函数来解决。然而，转换为这些间接优化通常会引入有效性损失。本文通过利用扩散过程的底层随机结构克服了这些基本障碍。与传统的影响目标类似，免疫目标是一个期望，可以表示为确定性实例上目标函数之和。但与前者不同，其中某些项并非子模。关键步骤在于证明该和函数与子模性的偏差有界，从而使贪心算法能够实现常数因子近似。我们证明，在考虑多种免疫设置和传播模型的情况下，该近似仍然成立。