  In the submodular cover problem, we are given a non-negative monotone
submodular function $f$ over a ground set $E$ of items, and the goal is to
choose a smallest subset $S \subseteq E$ such that $f(S) = Q$ where $Q = f(E)$.
In the stochastic version of the problem, we are given $m$ stochastic items
which are different random variables that independently realize to some item in
$E$, and the goal is to find a smallest set of stochastic items whose
realization $R$ satisfies $f(R) = Q$. The problem captures as a special case
the stochastic set cover problem and more generally, stochastic covering
integer programs.
  We define an $r$-round adaptive algorithm to be an algorithm that chooses a
permutation of all available items in each round $k \in [r]$, and a threshold
$\tau_k$, and realizes items in the order specified by the permutation until
the function value is at least $\tau_k$. The permutation for each round $k$ is
chosen adaptively based on the realization in the previous rounds, but the
ordering inside each round remains fixed regardless of the realizations seen
inside the round. Our main result is that for any integer $r$, there exists a
poly-time $r$-round adaptive algorithm for stochastic submodular cover whose
expected cost is $\tilde{O}(Q^{{1}/{r}})$ times the expected cost of a fully
adaptive algorithm. Prior to our work, such a result was not known even for the
case of $r=1$ and when $f$ is the coverage function. On the other hand, we show
that for any $r$, there exist instances of the stochastic submodular cover
problem where no $r$-round adaptive algorithm can achieve better than
$\Omega(Q^{{1}/{r}})$ approximation to the expected cost of a fully adaptive
algorithm. Our lower bound result holds even for coverage function and for
algorithms with unbounded computational power.
