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：本文定义了一类新的集合函数，这类函数除具有单调性和次可加性外，还在基集合的某个排列上具有非常受限的子模性，我们称该排列为子模序。这类函数包含单调子模函数作为其子类。我们给出了在多种约束条件下最大化子模序函数的快速算法，具有强近似保证，并证明了多项式查询复杂度算法所能达到的最高近似保证存在几乎紧的上界。将该新概念应用于基本选择模型中受约束的品类优化问题，我们获得了同时具有更快速度和更强近似保证的新算法（在某些情况下，这是首个具有常数因子保证的算法）。我们还展示了其与流式模型中单调子模函数最大化之间的有趣联系，作为我们结果的推论，恢复出已知的最佳近似保证。