We define a new class of set functions that in addition to being monotone and subadditive, also admit a very limited form of submodularity defined over a permutation of the ground set. We refer to this permutation as a submodular order. This class of functions includes monotone submodular functions as a sub-family. We give fast algorithms with strong approximation guarantees for maximizing submodular order functions under a variety of constraints and show a nearly tight upper bound on the highest approximation guarantee achievable by algorithms with polynomial query complexity. Applying this new notion to the problem of constrained assortment optimization in fundamental choice models, we obtain new algorithms that are both faster and have stronger approximation guarantees (in some cases, first algorithm with constant factor guarantee). We also show an intriguing connection to the maximization of monotone submodular functions in the streaming model, where we recover best known approximation guarantees as a corollary of our results.

翻译：我们定义了一类新的集合函数，这类函数在具备单调性和次可加性的同时，还在基集的一个排列上允许一种非常受限的子模性形式。我们将该排列称为子模序。这类函数包含单调子模函数作为其子族。我们提出了在多种约束下最大化子模序函数的快速算法，并具有强近似保证，同时证明了在多项式查询复杂度的算法中，近似保证的上界几乎是紧的。将这一新概念应用于基本选择模型中的受约束品类优化问题，我们得到了新算法，这些算法不仅速度更快，而且具有更强的近似保证（在某些情况下，是首个具有常数因子保证的算法）。我们还展示了与流式模型中单调子模函数最大化之间的有趣联系，并作为我们结果的推论，恢复了已知的最佳近似保证。