  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.
