Stochastic Multi-round Submodular Optimization with Budget

In this work we study the problem of Stochastic Budgeted Multi-round Submodular Maximization (SBMSm), in which we would like to maximize the sum over multiple rounds of the value of a monotone and submodular objective function, subject to the fact that the values of this function depend on the realization of stochastic events and the number of observations that we can make over all rounds is limited by a given budget. This problem extends, and generalizes to multiple round settings, well-studied problems such as (adaptive) influence maximization and stochastic probing. We first show that whenever a certain single-round optimization problem can be optimally solved in polynomial time, then there is a polynomial time dynamic programming algorithm that returns the same solution as the optimal algorithm, that can adaptively choose both which observations to make and in which round to have them. Unfortunately, this dynamic programming approach cannot be extended to work when the single-round optimization problem cannot be efficiently solved (even if we allow it would be approximated within an arbitrary small constant). Anyway, in this case we are able to provide a simple greedy algorithm for the problem. It guarantees a $(1/2-\epsilon)$-approximation to the optimal value, even if it non-adaptively allocates the budget to rounds.