  In the influence maximization (IM) problem, we are given a social network and
a budget $k$, and we look for a set of $k$ nodes in the network, called seeds,
that maximize the expected number of nodes that are reached by an influence
cascade generated by the seeds, according to some stochastic model for
influence diffusion. In this paper, we study the adaptive IM, where the nodes
are selected sequentially one by one, and the decision on the $i$th seed can be
based on the observed cascade produced by the first $i-1$ seeds. We focus on
the full-adoption feedback in which we can observe the entire cascade of each
previously selected seed and on the independent cascade model where each edge
is associated with an independent probability of diffusing influence.
  Our main result is the first sub-linear upper bound that holds for any graph.
Specifically, we show that the adaptivity gap is upper-bounded by $\lceil
n^{1/3}\rceil $, where $n$ is the number of nodes in the graph. Moreover, we
improve over the known upper bound for in-arborescences from
$\frac{2e}{e-1}\approx 3.16$ to $\frac{2e^2}{e^2-1}\approx 2.31$. Finally, we
study $\alpha$-bounded graphs, a class of undirected graphs in which the sum of
node degrees higher than two is at most $\alpha$, and show that the adaptivity
gap is upper-bounded by $\sqrt{\alpha}+O(1)$. Moreover, we show that in
0-bounded graphs, i.e. undirected graphs in which each connected component is a
path or a cycle, the adaptivity gap is at most $\frac{3e^3}{e^3-1}\approx
3.16$. To prove our bounds, we introduce new techniques to relate adaptive
policies with non-adaptive ones that might be of their own interest.
