We consider the influence maximization problem over a temporal graph, where there is a single fixed source. We deviate from the standard model of influence maximization, where the goal is to choose the set of most influential vertices. Instead, in our model we are given a fixed vertex, or source, and the goal is to find the best time steps to transmit so that the influence of this vertex is maximized. We frame this problem as a spreading process that follows a variant of the susceptible-infected-susceptible (SIS) model and we focus on four objective functions. In the MaxSpread objective, the goal is to maximize the total number of vertices that get infected at least once. In the MaxViral objective, the goal is to maximize the number of vertices that are infected at the same time step. In the MaxViralTstep objective, the goal is to maximize the number of vertices that are infected at a given time step. Finally, in MinNonViralTime, the goal is to maximize the total number of vertices that get infected every $d$ time steps. We perform a thorough complexity theoretic analysis for these four objectives over three different scenarios: (1) the unconstrained setting where the source can transmit whenever it wants; (2) the window-constrained setting where the source has to transmit at either a predetermined, or a shifting window; (3) the periodic setting where the temporal graph has a small period. We prove that all of these problems, with the exception of MaxSpread for periodic graphs, are intractable even for very simple underlying graphs.

翻译：我们考虑时态图上的影响力最大化问题，其中存在一个固定源。我们偏离标准的影响力最大化模型（其目标是选择最具影响力的顶点集合），转而研究给定固定顶点（源）的情况下，如何选择最佳传播时间步以最大化该顶点的影响力。我们将此问题建模为遵循易感-感染-易感（SIS）模型变体的传播过程，并聚焦于四种目标函数。在MaxSpread目标中，目标是最大化至少被感染一次的顶点总数。在MaxViral目标中，目标是最大化在同一时间步被感染的顶点数量。在MaxViralTstep目标中，目标是最大化在给定时间步被感染的顶点数量。最后，在MinNonViralTime目标中，目标是最大化每$d$个时间步被感染的顶点总数。我们对这四种目标在三种不同场景下进行了深入的复杂性理论分析：（1）无约束设置，即源可在任意时刻传播；（2）窗口约束设置，即源必须在预定或滑动窗口内传播；（3）周期设置，即时态图具有小周期。我们证明，除周期性图上的MaxSpread问题外，其他所有问题即便在极其简单的底层图结构上也是难以处理的。