  In this paper, we propose the amphibious influence maximization (AIM) model
that combines traditional marketing via content providers and viral marketing
to consumers in social networks in a single framework. In AIM, a set of content
providers and consumers form a bipartite network while consumers also form
their social network, and influence propagates from the content providers to
consumers and among consumers in the social network following the independent
cascade model. An advertiser needs to select a subset of seed content providers
and a subset of seed consumers, such that the influence from the seed providers
passing through the seed consumers could reach a large number of consumers in
the social network in expectation.
  We prove that the AIM problem is NP-hard to approximate to within any
constant factor via a reduction from Feige's k-prover proof system for 3-SAT5.
We also give evidence that even when the social network graph is trivial (i.e.
has no edges), a polynomial time constant factor approximation for AIM is
unlikely. However, when we assume that the weighted bi-adjacency matrix that
describes the influence of content providers on consumers is of constant rank,
a common assumption often used in recommender systems, we provide a
polynomial-time algorithm that achieves approximation ratio of
$(1-1/e-\epsilon)^3$ for any (polynomially small) $\epsilon > 0$. Our
algorithmic results still hold for a more general model where cascades in
social network follow a general monotone and submodular function.
