Minimize sum of array sample from columns in 2D array with limitation on number of used row

Question

TLDR; It seems to be a top k problem with dependency. Or a Knapsack Problem but the order matters.

If I have a 2D array with (N, M) shape. I need to select L rows from it to be a (L, M) shape array SUB_M. And for each column in SUB_M. I pick the smallest value in it and become a 1D (M) shape array ANS_M My goal is to minimize the sum of ANS_M.

So for array like this.

[[  8   7   7   7   7   7   7   5   6]
 [  5   7   6   5   6   6   7   7   5]
 [  7   8   8   8   5   5   5   6   7]
 [  7   8   7   7   7   6   7   8   5]
 [  0   0 100 100 100 100 100 100 100]
 [100 100 100   0   0 100 100 100 100]
 [100 100 100 100 100 100   0   0 100]]

I'll select row 1, 4, 5, 6. The SUB_M would looks like this:

[[  5   7   6   5   6   6   7   7   5]
 [  0   0 100 100 100 100 100 100 100]
 [100 100 100   0   0 100 100 100 100]
 [100 100 100 100 100 100   0   0 100]]

And the ANS_M is this:

[  0   0   6   0   0   6   0   0   5]

So the sum of ANS_M is 17.

I think this problem is a little bit like a k sum problem. But unlike k-sum problem. This problem can't be solved using max/min heap. Because in k-sum problem, two number can be compare directly. But in my problem, even if arr A is smaller than B and C in sum. Maybe sum of the combination of B and C can be smaller than A. For example.

A = [ 5  5  5  5  5]
B = [ 0  0  0 99 99]
C = [99 99 99  0  0]

I was banging my head on wall on the problem. And I can't make ChatGPT to solve it.

I have a brute force solution is find all the combination C(N, C). And for each combination arrays, I find the minimum along N and iterating through all the combination.

Is there any better way? Because the N and M could be pretty large.