Maximizing Monotone Submodular Functions over the Integer Lattice

The problem of maximizing non-negative monotone submodular functions under a certain constraint has been intensively studied in the last decade. In this paper, we address the problem for functions defined over the integer lattice. Suppose that a non-negative monotone submodular function $f:\mathbb{Z}+ⁿ \to \mathbb{R}+$ is given via an evaluation oracle. Assume further that $f$ satisfies the diminishing return property, which is not an immediate consequence of submodularity when the domain is the integer lattice. Given this, we design polynomial-time $(1-1/e-\epsilon)$-approximation algorithms for a cardinality constraint, a polymatroid constraint, and a knapsack constraint. For a cardinality constraint, we also provide a $(1-1/e-\epsilon)$-approximation algorithm with slightly worse time complexity that does not rely on the diminishing return property.