We consider a problem of maximizing a monotone DR-submodular function under multiple order-consistent knapsack constraints on a distributive lattice. Because a distributive lattice is used to represent a dependency constraint, the problem can represent a dependency constrained version of a submodular maximization problem on a set. We propose a (\(1 - 1/e\))-approximation algorithm for this problem. To achieve this result, we generalize the continuous greedy algorithm to distributive lattices: We choose a median complex as a continuous relaxation of the distributive lattice and define the multilinear extension on it. We show that the median complex admits special curves, named uniform linear motions. The multilinear extension of a DR-submodular function is concave along a positive uniform linear motion, which is a key property used in the continuous greedy algorithm.

我们考虑在分配格上的多个顺序一致的背包约束下最大化单调DR子模函数的问题。因为使用分布格来表示依赖关系约束，所以该问题可以表示集合上的子模块化最大化问题的依赖关系约束版本。我们针对此问题提出（\（1/1 / e \））逼近算法。为了获得此结果，我们将连续贪婪算法推广到分布晶格：我们选择中位数复合体作为分布晶格的连续松弛，并在其上定义多线性扩展。我们表明，中位数复数接受特殊曲线，称为均匀线性运动。DR子模函数的多线性扩展是沿着正均匀线性运动凹入的，这是连续贪婪算法中使用的关键属性。