  The typical algorithmic problem in viral marketing aims to identify a set of
influential users in a social network, who, when convinced to adopt a product,
shall influence other users in the network and trigger a large cascade of
adoptions. However, the host (the owner of an online social platform) often
faces more constraints than a single product, endless user attentions,
unlimited budget and unbounded time; in reality, multiple products need to be
advertised, each user can tolerate only a small number of recommendations,
influencing user has a cost and advertisers have only limited budgets, and the
adoptions need to be maximized within a short time window.
  Given theses myriads of user, monetary, and timing constraints, it is
extremely challenging for the host to design principled and efficient viral
market algorithms with provable guarantees. In this paper, we provide a novel
solution by formulating the problem as a submodular maximization in a
continuous-time diffusion model under an intersection of a matroid and multiple
knapsack constraints. We also propose an adaptive threshold greedy algorithm
which can be faster than the traditional greedy algorithm with lazy evaluation,
and scalable to networks with million of nodes. Furthermore, our mathematical
formulation allows us to prove that the algorithm can achieve an approximation
factor of $k_a/(2+2 k)$ when $k_a$ out of the $k$ knapsack constraints are
active, which also improves over previous guarantees from combinatorial
optimization literature. In the case when influencing each user has uniform
cost, the approximation becomes even better to a factor of $1/3$. Extensive
synthetic and real world experiments demonstrate that our budgeted influence
maximization algorithm achieves the-state-of-the-art in terms of both
effectiveness and scalability, often beating the next best by significant
margins.
