Maximization Problem for Independence Systems

Problem
Let $S$ be a finite set.

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

Let $\mathscr F$ be an independence system 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 $w$.

Also see

 * Greedy Algorithm yields Maximal Set where it is shown that the output from the greedy algorithm for the Maximum Weight Problem $\struct{\mathscr F, w}$ is a maximal set of $\mathscr F$.


 * Greedy Algorithm may not yield Maximum Weight where it is shown that the maximal set of $\mathscr F$ constructed by the greedy algorithm is not guaranteed to be of maximum weight.


 * Greedy Algorithm guarantees Maximum Weight iff Matroid where it can be seen that the greedy algorithm only guarantees that the chosen maximal set has maximum weight iff $\struct{S, \mathscr F}$ is a matroid.