Bomb dropping algorithm

Question

I have an n x m matrix consisting of non-negative integers. For example:

2 3 4 7 1
1 5 2 6 2
4 3 4 2 1
2 1 2 4 1
3 1 3 4 1
2 1 4 3 2
6 9 1 6 4
         


        

Answer 1

It seems that a linear programming approach can be very helpful here.

Let P_{m x n} be the matrix with the values of the positions:

Now let define a bomb matrix B(x, y)_{m x n},with 1 ≤ x ≤ m, 1 ≤ y ≤ n as below

in such a way that

For example:

So we are looking to a matrix B_{m x n} = [b_ij] that

1. Can be defined as a sum of bomb matrices:

   

   (q_ij would be then the quantity of bombs we would drop in position p_ij)

2. p_ij - b_ij ≤ 0 (to be more succint, let us say it as P - B ≤ 0)

Also, B should minimize the sum .

We can also write B as the ugly matrix ahead:

and since P - B ≤ 0 (which means P ≤ B) we have the following pretty linear inequality system below:

Being q_{mn x 1} defined as

p_{mn x 1} defined as

We can say we have a system The system below represented as product of matrices http://latex.codecogs.com/gif.download?S%5Cmathbf%7Bq%7D&space;%5Cge&space;%5Cmathbf%7Bp%7D being S_{mn x mn} the matrix to be reversed to solve the system. I did not expand it myself but I believe it should be easy to do it in code.

Now, we have a minimum problem which can be stated as

I believe it is something easy, almost trivial to be solved with something like the simplex algorithm (there is this rather cool doc about it). However, I do know almost no linear programming (I will take a course about it on Coursera but it is just in the future...), I had some headaches trying to understand it and I have a huge freelance job to finish so I just give up here. It can be that I did something wrong at some point, or that it can't go any further, but I believe this path can eventually lead to the solution. Anyway, I am anxious for your feedback.

(Special thanks for this amazing site to create pictures from LaTeX expressions)