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We consider a problem of maximizing a monotone DR-submodular function undermultiple order-consistent knapsack constraints on a distributive lattice. Sincea distributive lattice is used to represent a dependency constraint, theproblem can represent a dependency constrained version of a submodularmaximization problem on a set. We propose a $1 - 1/e$ approximation algorithmfor this problem. To achieve this result, we generalize the continuous greedyalgorithm to distributive lattices: We choose a median complex as a continuousrelaxation of a distributive lattice and define the multilinear extension onit. We show that the median complex admits special curves, named uniform linearmotions, such that the multilinear extension of a DR-submodular function isconcave along a positive uniform linear motion, which is a key property of thecontinuous greedy algorithm.

Multiple Knapsack-Constrained Monotone DR-Submodular Maximization on Distributive Lattice --- Continuous Greedy Algorithm on Median Complex ---