  We demonstrate that from an algorithm guaranteeing an approximation factor
for the ratio of submodular (RS) optimization problem, we can build another
algorithm having a different kind of approximation guarantee -- weaker than the
classical one -- for the difference of submodular (DS) optimization problem,
and vice versa. We also illustrate the link between these two problems by
analyzing a \textsc{Greedy} algorithm which approximately maximizes objective
functions of the form $\Psi(f,g)$, where $f,g$ are two non-negative, monotone,
submodular functions and $\Psi$ is a {quasiconvex} 2-variables function, which
is non decreasing with respect to the first variable. For the choice
$\Psi(f,g)\triangleq f/g$, we recover RS, and for the choice
$\Psi(f,g)\triangleq f-g$, we recover DS. To the best of our knowledge, this
greedy approach is new for DS optimization. For RS optimization, it reduces to
the standard \textsc{GreedRatio} algorithm that has already been analyzed
previously. However, our analysis is novel for this case.

Data Structures and Algorithms