A Simple Extreme Subsets Algorithm for Submodular and Posi-modular Set Functions

Let f be a set function on a finite set V . A pair {u, v} ⊆ V is called flat in f if f(X) ≥ minx∈X f({x}) holds for all subsets X ⊆ V with |X ∩ {u, v}| = 1. A subset X ⊆ V is called an extreme subset of f if f(Y ) > f(X) holds for all nonempty and proper subsets Y of X. In this paper, we first define a minimum degree ordering (MD ordering) of V , and show that the last two elements in an MD ordering give a flat pair of f if f is symmetric and crossing submodular. Based on this, we then give a simple O(|V |3)-oracle time algorithm for computing all extreme subsets of a set function g if g is symmetric and crossing submodular or intersecting submodular and posi-modular.