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We demonstrate that from an algorithm guaranteeing an approximation factorfor the ratio of submodular (RS) optimization problem, we can build anotheralgorithm having a different kind of approximation guarantee -- weaker than theclassical one -- for the difference of submodular (DS) optimization problem,and vice versa. We also illustrate the link between these two problems byanalyzing a \textsc{Greedy} algorithm which approximately maximizes objectivefunctions of the form $\Psi(f,g)$, where $f,g$ are two non-negative, monotone,submodular functions and $\Psi$ is a {quasiconvex} 2-variables function, whichis 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, thisgreedy approach is new for DS optimization. For RS optimization, it reduces tothe standard \textsc{GreedRatio} algorithm that has already been analyzedpreviously. However, our analysis is novel for this case.

On the Approximation Relationship between Optimizing Ratio of Submodular (RS) and Difference of Submodular (DS) Functions