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 cannot be solved in polynomial time 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，通过仅控制少量变量的取值，最大化$+1$预期数量的最优方法是什么？这类问题被称为影响力最大化问题，已在社交网络、生物学和计算机科学中得到广泛研究。本文考虑无向图模型典型代表——伊辛模型上的影响力最大化问题。该模型在众多实际应用问题中具有广泛适用性。我们在有界预算约束下，揭示了稀疏伊辛模型上影响力最大化的严格计算相变：在高温区域，我们提出一种线性时间算法，能够找到一组小规模变量及其取值，实现近乎最优的影响力；在低温区域，我们证明在公认的计算复杂性假设下，该问题无法在多项式时间内求解。临界温度恰好对应于伊辛模型的树状唯一性/非唯一性阈值，该阈值也是近似采样与计数等其他计算问题的临界点。