  We study the adaptive influence maximization problem with myopic feedback
under the independent cascade model: one sequentially selects k nodes as seeds
one by one from a social network, and each selected seed returns the immediate
neighbors it activates as the feedback available for later selections, and the
goal is to maximize the expected number of total activated nodes, referred as
the influence spread. We show that the adaptivity gap, the ratio between the
optimal adaptive influence spread and the optimal non-adaptive influence
spread, is at most 4 and at least e/(e-1), and the approximation ratios with
respect to the optimal adaptive influence spread of both the non-adaptive
greedy and adaptive greedy algorithms are at least \frac{1}{4}(1 - \frac{1}{e})
and at most \frac{e^2 + 1}{(e + 1)^2} < 1 - \frac{1}{e}. Moreover, the
approximation ratio of the non-adaptive greedy algorithm is no worse than that
of the adaptive greedy algorithm, when considering all graphs. Our result
confirms a long-standing open conjecture of Golovin and Krause (2011) on the
constant approximation ratio of adaptive greedy with myopic feedback, and it
also suggests that adaptive greedy may not bring much benefit under myopic
feedback.
