We consider the \emph{adaptive influence maximization problem}: given a
network and a budget k, iteratively select k seeds in the network to
maximize the expected number of adopters. In the \emph{full-adoption feedback
model}, after selecting each seed, the seed-picker observes all the resulting
adoptions. In the \emph{myopic feedback model}, the seed-picker only observes
whether each neighbor of the chosen seed adopts. Motivated by the extreme
success of greedy-based algorithms/heuristics for influence maximization, we
propose the concept of \emph{greedy adaptivity gap}, which compares the
performance of the adaptive greedy algorithm to its non-adaptive counterpart.
Our first result shows that, for submodular influence maximization, the
adaptive greedy algorithm can perform up to a (1β1/e)-fraction worse than the
non-adaptive greedy algorithm, and that this ratio is tight. More specifically,
on one side we provide examples where the performance of the adaptive greedy
algorithm is only a (1β1/e) fraction of the performance of the non-adaptive
greedy algorithm in four settings: for both feedback models and both the
\emph{independent cascade model} and the \emph{linear threshold model}. On the
other side, we prove that in any submodular cascade, the adaptive greedy
algorithm always outputs a (1β1/e)-approximation to the expected number of
adoptions in the optimal non-adaptive seed choice. Our second result shows
that, for the general submodular cascade model with full-adoption feedback, the
adaptive greedy algorithm can outperform the non-adaptive greedy algorithm by
an unbounded factor. Finally, we propose a risk-free variant of the adaptive
greedy algorithm that always performs no worse than the non-adaptive greedy
algorithm.Comment: 26 pages, 0 figure, accepted at AAAI'20: Thirty-Fourth AAAI
Conference on Artificial Intelligenc