We have an $n \times m$ matrix whose entries are non-negative integers and we want to find a sub-matrix whose area (number of entries) is at least $k$ such that the sum of the entries in minimal. The answer to this problem is the sum of these entries.
My solution: Using dynamic programming, I have defined the sum of entries from the (0,0) entry to (i,j) as $sum[i][j]$. If the $(i,j)$ entry is $entry[i][j]$ then we have $sum[i][j] = sum[i-1][j] + sum[i][j-1] - sum[i-1][j-1] + entry[i][j]$. Except, when either dimension of $sum$ is 0, the values are only the entries.
I do not know how to continue from here.
If you have a better idea for this problem, that would be appreciated too!