Abstract

We investigate a new optimization problem involving minimizing the Ratio of two Submodular (RS) functions. We argue that this problem occurs naturally in several real world applications We then show the connection between this problem and several related problems including minimizing the difference between submodular functions (Iyer & Bilmes, 2012b; Narasimhan & Bilmes, 2005), and to submodular optimization subject to submodular constraints (Iyer & Bilmes, 2013). We show that RS optimization can be solved with bounded approximation factors. We also provide a hardness bound and show that our tightest algorithm matches the lower bound up to a log factor. Finally, we empirically demonstrate the performance and good scalability properties of our algorithms.