Notes on Greedy Algorithms for Submodular Maximization

All the functions we consider are set functions defined over subsets of a ground set N . Definition 1. A function f : 2 → R is monotone iff: ∀S ⊆ T ⊆ N, f(S) ≤ f(T ) Definition 2. For f : 2 → R and S ⊆ N , the marginal contribution to S is the function fS defined by: ∀T ⊆ N, fS(T ) = f(S ∪ T )− f(S) When there is no ambiguity, we write fS(e) instead of fS({e}) for e ∈ N , S + e instead of S ∪ {e} and S − e instead of S \ {e}. Definition 3. A function f : 2 → R is submodular iff: ∀S ⊆ T ⊆ N, ∀e ∈ N \ T, fT (e) ≤ fS(e) This “decreasing marginal contribution” definition of submodular functions often leads to treating them as “discrete concave functions”. Proposition 4. The following statements are equivalent: 1. f is submodular. 2. for all S ⊆ N , fS is submodular. 3. for all S ⊆ N , fS is subadditive. Proof. (1. ⇒ 2.) is immediate. To prove (2. ⇒ 3.), we show that any submodular function f is subadditive. Let f be a submodular function. Consider A and B two subets of N . Writing B = {e1, . . . , en} and Bi = {e1, . . . , ei}: