Given a complex high-dimensional distribution over $\{\pm 1\}^n$, what is the best way to increase the expected number of $+1$'s by controlling the values of only a small number of variables? Such a problem is known as influence maximization and has been widely studied in social networks, biology, and computer science. In this paper, we consider influence maximization on the Ising model which is a prototypical example of undirected graphical models and has wide applications in many real-world problems. We establish a sharp computational phase transition for influence maximization on sparse Ising models under a bounded budget: In the high-temperature regime, we give a linear-time algorithm for finding a small subset of variables and their values which achieve nearly optimal influence; In the low-temperature regime, we show that the influence maximization problem becomes $\mathsf{NP}$-hard under commonly-believed complexity assumption. The critical temperature coincides with the tree uniqueness/non-uniqueness threshold for Ising models which is also a critical point for other computational problems including approximate sampling and counting.

翻译：给定一个定义在$\{\pm 1\}^n$上的复杂高维分布，如何通过仅控制少量变量的取值来最大化$+1$数量的期望值？这类问题被称为影响力最大化，已在社交网络、生物学和计算机科学领域得到广泛研究。本文考虑无向图模型典型范例——伊辛模型上的影响力最大化问题，该模型在众多实际问题中具有广泛应用。我们揭示了有限预算下稀疏伊辛模型影响力最大化的精确计算相变：在高温区域，我们提出一种线性时间算法，能够找到接近最优影响力的变量子集及其对应赋值；在低温区域，我们证明在公认的复杂性假设下影响力最大化问题属于$\mathsf{NP}$-困难问题。该临界温度与伊辛模型的树状唯一性/非唯一性阈值一致，该阈值同时也是近似采样与计数等其他计算问题的关键临界点。