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

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 textscGreedy 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 textscGreedRatio algorithm that has already been analyzed previously. However, our analysis is novel for this case.