Overcoming Non-Submodularity: Towards Constant Approximation for Network Immunization

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.