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

## TL;DR

This paper extends the continuous greedy algorithm to distributive lattices for maximizing monotone DR-submodular functions under multiple knapsack constraints, providing a $1 - 1/e$ approximation.

## Contribution

It introduces a novel median complex relaxation and proves concavity of the multilinear extension along specific curves, enabling the generalized continuous greedy algorithm.

## Key findings

- Achieves a $1 - 1/e$ approximation ratio.
- Generalizes continuous greedy algorithm to distributive lattices.
- Introduces median complex as a continuous relaxation.

## Abstract

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.