A short proof for stronger version of DS decomposition in set function optimization

Abstract

Using a short proof, we show that every set function f can be decomposed into the difference of two monotone increasing and strictly submodular functions g and h, i.e., $$f=g-h$$ , and every set function f can also be decomposed into the difference of two monotone increasing and strictly supermodular functions g and h, i.e., $$f=g-h$$ .