I’m a bit confused regarding the following question:
Let $m,n \geqslant2$ be integers and $M=[1,2,....,m],N=[1,2,...,n]$. I’ve shown that the system $S=(E,J)$ is a matroid where $E=M$x$N$ (the set of cells of a rectangular matrix) and $I\in J$ if and only if $I$ is a set of cells where no three belong to the same column using the Exchange Proposition. I’ve then formulated the combinatorial optimisation problem associated with S. However, the part I’m struggling with is using GREEDY to solve the COP I formulated previously for the matrix $$A=\begin{pmatrix} 9 & 13& 3 & 12&2 \\ 10& 11 & 0 & 13 &1 \\ 12 & 7 &1 & 9 & 3 \end{pmatrix}$$ The reason I’m confused is I know using GREEDY that I pick the most attractive cell i.e. $0$ and then I would reject all enteries in the same row and column and then repeat. Doing this I get, what I think is the optimal cost to be $9$. However I believe (I could be mistaken as we haven’t covered this method) that I could solve this by using something called the Hungarian method in which case the optimal cost is $38$. I’m just really confused if I’m being honest because I don’t understand how I use GREEDY to solve the COP problem formulated, so any help would be appreciated.