Suppose we have a matrix with $R$ rows and $C$ columns.
1.An element appears no more than once in the same row.
2.The set of elements in each row has intersection.
3.Elements in the matrix can only be moved in the same row.
Objective: Through the movements of the elements in the same row, get the minimum sum of the numbers of unique elements in each column.
$\min(\sum_{i=1}^Cunique(column_i))$
Example: $$\begin {bmatrix} {a}&{d}&{f}&{b}\\ {d}&{e}&{c}&{g}\\ {b}&{a}&{e}&{s}\\ {s}&{b}&{a}&{h}\\ \end{bmatrix}$$
Objective = 4+4+4+4=16
After movements:
$$\begin {bmatrix} {a}&{b}&{d}&{f}\\ {c}&{g}&{d}&{e}\\ {a}&{b}&{s}&{e}\\ {a}&{b}&{s}&{f}\\ \end{bmatrix}$$
Objective = 2+2+2+2 = 8
This is a example only with 4 columns and 4 rows.what about a matrix with many columns and rows? How to find the optimal movements policy and prove it?
Thank you!
