I am working on the following exercise:
Let $(S,\mathcal{F})$ be a matroid and let $c:S \rightarrow \mathbb{R}$ be a weight function on $S$. Find an algorithm that solves the following problem:
MAXIMISATION PROBLEM FOR INDEPENDENCE SYSTEMS (MAX IS): Find an $F \in \mathcal{F}$ such that $c(F)$ is maximal.
From the lecture I know that the Greedy Algorithm associated with matroids can solve the similar problem:
MINIMISATION PROBLEM FOR INDEPENDENCE SYSTEMS (MIN IS): Find a basis $B$ in $\mathcal{F}$ such that $c(B)$ is minimal.
So the question is how we can relate these two problems with one another. If we were considering minimum spanning trees I would simply change the signs of $c$ and use Kruskal's algorithm. However, it confuses me that MIN IS works over bases, but on the other hand I do realise that the $F$ from MAX IS have to be bases in case of positive weights...
Is there a way to make this connection here?
