Maximization Problem for Independence Systems

Definition
Let $S$ be a finite set.

Let $w:S \to \R_{\ge 0}$ be a weight function.

Let $\mathscr F$ be a hereditary set of subsets of $S$.

The maximum weight problem $\struct{\mathscr F, w}$ is the computational problem to find a subset $A_0$ of $S$ with the properties:
 * $(1)\quad A_0 \in \mathscr F$
 * $(2)\quad \map {w^+} {A_0} = \max \set{\map {w^+} A : A \in \mathscr F}$

where $w^+$ is the extended weight function of $w$.