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.

翻译：在本文中，我们研究子模优化中的一个基本问题，即序列子模最大化。具体而言，我们旨在从基础集合 $V$ 中选择并排序一组 $k$ 个物品，使得 $k$ 个（可能非单调的）子模函数 $f_1, \cdots ,f_k: 2^V \rightarrow \mathbb{R}^+$ 的加权和最大化，其中每个函数 $f_j$ 以该序列中的前 $j$ 个物品作为输入。现有关于序列子模最大化的研究主要集中在单调设定上，即假设子模函数是非递减的。然而，在多样化推荐系统等现实场景中，向现有集合添加物品可能会对整体效用产生负面影响。为此，本文首次探讨了上述问题在非单调子模函数下的情形，并为灵活长度约束和固定长度约束，以及具备相同效用函数的特例提供了有效解决方案。实证评估进一步验证了我们提出的算法在视频推荐领域的有效性。本研究的结果对包括推荐系统和品类优化在内的多个领域具有启示意义，其中物品的排序会显著影响所获得的整体价值。