  The abstract model of stochastic probing was presented by Gupta and Nagarajan
(IPCO'13), and provides a unified view of a number of problems. Adamczyk,
Sviridenko, Ward (STACS'14) gave better approximation for matroid environments
and linear objectives. At the same time this method was easily extendable to
settings, where the objective function was monotone submodular. However, the
case of non-negative submodular function could not be handled by previous
techniques. In this paper we address this problem, and our results are twofold.
First, we adapt the notion of contention resolution schemes of Chekuri,
Vondr\'ak, Zenklusen (SICOMP'14) to show that we can optimize non-negative
submodular functions in this setting with a constant factor loss with respect
to the deterministic setting. Second, we show a new contention resolution
scheme for transversal matroids, which yields better approximations in the
stochastic probing setting than the previously known tools. The rounding
procedure underlying the scheme can be of independent interest --- Bansal,
Gupta, Li, Mestre, Nagarajan, Rudra (Algorithmica'12) gave two seemingly
different algorithms for stochastic matching and stochastic $k$-set packing
problems with two different analyses, but we show that our single technique can
be used to analyze both their algorithms.
