  We consider the maximization problem in the value oracle model of functions
defined on $k$-tuples of sets that are submodular in every orthant and $r$-wise
monotone, where $k\geq 2$ and $1\leq r\leq k$. We give an analysis of a
deterministic greedy algorithm that shows that any such function can be
approximated to a factor of $1/(1+r)$. For $r=k$, we give an analysis of a
randomised greedy algorithm that shows that any such function can be
approximated to a factor of $1/(1+\sqrt{k/2})$.
  In the case of $k=r=2$, the considered functions correspond precisely to
bisubmodular functions, in which case we obtain an approximation guarantee of
$1/2$. We show that, as in the case of submodular functions, this result is the
best possible in both the value query model, and under the assumption that
$NP\neq RP$.
  Extending a result of Ando et al., we show that for any $k\geq 3$
submodularity in every orthant and pairwise monotonicity (i.e. $r=2$) precisely
characterize $k$-submodular functions. Consequently, we obtain an approximation
guarantee of $1/3$ (and thus independent of $k$) for the maximization problem
of $k$-submodular functions.
