  Submodular maximization with a cardinality constraint can model various
problems, and those problems are often very large in practice. For the case
where objective functions are monotone, many fast approximation algorithms have
been developed. The stochastic greedy algorithm (SG) is one such algorithm,
which is widely used thanks to its simplicity, efficiency, and high empirical
performance. However, its approximation guarantee has been proved only for
monotone objective functions. When it comes to non-monotone objective
functions, existing approximation algorithms are inefficient relative to the
fast algorithms developed for the case of monotone objectives. In this paper,
we prove that SG (with slight modification) can achieve almost
$1/4$-approximation guarantees in expectation in linear time even if objective
functions are non-monotone. Our result provides a constant-factor approximation
algorithm with the fewest oracle queries for non-monotone submodular
maximization with a cardinality constraint. Experiments validate the
performance of (modified) SG.
