  We consider a problem of maximizing a monotone DR-submodular function under
multiple order-consistent knapsack constraints on a distributive lattice. Since
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 a distributive lattice and define the multilinear extension on
it. We show that the median complex admits special curves, named uniform linear
motions, such that the multilinear extension of a DR-submodular function is
concave along a positive uniform linear motion, which is a key property of the
continuous greedy algorithm.
