  In this paper we consider the problem of maximizing a non-negative submodular
function subject to a cardinality constraint in the data stream model.
Previously, the best known algorithm for this problem was a
$5.828$-approximation semi-streaming algorithm based on a local search
technique (Feldman et al., 2018). For the special case of this problem in which
the objective function is also monotone, the state-of-the-art semi-streaming
algorithm is an algorithm known as Sieve-Streaming, which is based on a
different technique (Badanidiyuru, 2014). Adapting the technique of
Sieve-Streaming to non-monotone objective functions has turned out to be a
challenging task, which has so far prevented an improvement over the local
search based $5.828$-approximation. In this work, we overcome the above
challenge, and manage to adapt Sieve-Streaming to non-monotone objective
functions by introducing a "just right" amount of randomness into it.
Consequently, we get a semi-streaming polynomial time $4.282$-approximation
algorithm for non-monotone objectives. Moreover, if one allows our algorithm to
run in super-polynomial time, then its approximation ratio can be further
improved to $3 + \varepsilon$.
