Marginal increment of a set function (${\Delta }_{x}f(S)=f(S\cup \{x\})-f(S)$)

The marginal increment measures how much a set function changes when one new element is added. Usually $x\notin S$ is required; if $x\in S$, the displayed difference is zero. The marginal increments of a modular set function depend only on $x$. The marginal increments of a submodular set function can only decrease as the input subset grows without acquiring $x$.