  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.
