This paper investigates the influence maximization problem under the Independent Cascade(IC) and Linear Threshold (LT) models. While this problem is known to be APX-hard on general graphs, we explore its computational limits by focusing on Directed Acyclic Graphs (DAGs) and more restricted tree structures. Our primary result demonstrates that influence maximization remains APX-hard on DAGs under the LT model, suggesting that the absence of cycles is insufficient to achieve a polynomial-time approximation scheme (PTAS). In contrast, we show that the problem becomes tractable when the topology is further restricted to out-arborescences and in-arborescences. Specifically, for out-arborescences, we show that the IC model and the LT model are equivalent, and we develop exact polynomial-time algorithms based on dynamic programming that leverage the unique path properties of these structures. For in-arborescences, it is known that the problem is polynomial-time solvable under the LT model, and it is NP-hard under the IC model. We complement these results by presenting a fully polynomial-time approximation scheme (FPTAS) for the IC model.

翻译：本文研究了独立级联（IC）模型与线性阈值（LT）模型下的影响力最大化问题。尽管该问题在一般图上已知为APX-hard，我们通过聚焦于有向无环图（DAG）及更受限的树形结构，探究其计算极限。我们的主要结果表明：在LT模型下，影响力最大化在DAG上仍保持APX-hard性，表明无环性不足以实现多项式时间近似方案（PTAS）。相反，我们证明当拓扑结构进一步限制为外向树与内向树时，该问题变得可解。具体而言，对于外向树，我们证明了IC模型与LT模型等价，并基于动态规划提出了精确的多项式时间算法，该算法利用此类结构的唯一路径特性。对于内向树，已知该问题在LT模型下可在多项式时间内求解，而在IC模型下为NP-hard。我们通过为IC模型提出一个完全多项式时间近似方案（FPTAS）来补充这些结果。