# Submodular function

Let V be a finite ground set. A set function $b:2^V \to {\mathbb R} \cup \{\infty\}$ is (fully) submodular if $b(X)+b(Y) \geq b(X \cap Y)+b(X \cup Y)$ for every $X,Y \subseteq V$. In other words, b is submodular if and only if -b is a supermodular function.
• p is intersecting submodular if the submodular inequality holds for every $X,Y \subseteq V$ for which $X \cap Y \neq \emptyset$.
• p is crossing submodular if the submodular inequality holds for every $X,Y \subseteq V$ for which $X \cap Y \neq \emptyset$ and $X \cup Y \neq V$.