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

TL;DR

This paper explores the theoretical relationship between ratio and difference of submodular functions, proposing a greedy algorithm that unifies their approximation strategies and analyzing its novelty and implications.

Contribution

It introduces a new greedy algorithm linking RS and DS optimization, providing novel analysis for both problem types and demonstrating their approximation relationship.

Findings

A greedy algorithm unifies RS and DS optimization approaches.

The analysis reveals a theoretical link between ratio and difference of submodular functions.

The approach offers new insights into approximation guarantees for both problems.

Abstract

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.