  The problem of maximizing non-negative submodular functions has been studied
extensively in the last few years. However, most papers consider submodular set
functions. Recently, several advances have been made for the more general case
of submodular functions on the integer lattice. In this paper, we present a
deterministic $\frac{1}{3}$-approximation for maximizing a submodular function
on a bounded integer lattice $\{0, \ldots, C\}^n$ using a Double Greedy
framework. Moreover, we show that the analysis is tight and that other ideas
used for approximating set functions cannot easily be extended. In contrast to
set functions, submodularity on the integer lattice does not imply the
so-called diminishing returns property. Assuming this property, it was shown
that many results for set functions can also be obtained for the integer
lattice. In this paper, we consider a further generalization. Instead of the
integer lattice, we consider a distributive lattice as the function domain and
assume the diminishing returns (DR) property. On the one hand, we show that
some approximation algorithms match the set functions setting. In particular,
we can obtain a $\frac{1}{2}$-approximation for unconstrained maximization, a
$(1-\frac{1}{e})$-approximation for monotone functions under a cardinality
constraint and a $\frac{1}{2}$-approximation for a poset matroid constraint. On
the other hand, for a knapsack constraint, the problem becomes significantly
harder: even for monotone DR-submodular functions, we show that there is no
$2^{(\log (n^{1/2} - 1))^\delta - 1}$-approximation for every $\delta > 0$
under the assumption that $3-SAT$ cannot be solved in time $2^{n^{3/4 +
\epsilon}}$.
