  In this paper, we study a fundamental problem in submodular optimization,
which is called sequential submodular maximization. Specifically, we aim to
select and rank a group of $k$ items from a ground set $V$ such that the
weighted summation of $k$ (possibly non-monotone) submodular functions $f_1,
\cdots ,f_k: 2^V \rightarrow \mathbb{R}^+$ is maximized, here each function
$f_j$ takes the first $j$ items from this sequence as input. The existing
research on sequential submodular maximization has predominantly concentrated
on the monotone setting, assuming that the submodular functions are
non-decreasing. However, in various real-world scenarios, like diversity-aware
recommendation systems, adding items to an existing set might negatively impact
the overall utility. In response, this paper pioneers the examination of the
aforementioned problem with non-monotone submodular functions and offers
effective solutions for both flexible and fixed length constraints, as well as
a special case with identical utility functions. The empirical evaluations
further validate the effectiveness of our proposed algorithms in the domain of
video recommendations. The results of this research have implications in
various fields, including recommendation systems and assortment optimization,
where the ordering of items significantly impacts the overall value obtained.
