Covering symmetric semi-monotone functions

We define a new set of functions called semi-monotone, a subclass of skew-supermodular functions. We show that the problem of augmenting a given graph to cover a symmetric semi-monotone function is NP-complete if all the values of the function are in image and we provide a minimax theorem if all the values of the function are different from 1. Our problem is equivalent to the node to area augmentation problem. Our contribution is to provide a significantly simpler and shorter proof.