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

Pólya says "If you can't solve a problem, then there is an easier problem you can solve: find it."

The obvious simpler problem is the 1-dimensional problem (when the grid is a single row). Let's start with the simplest algorithm - greedily bombing the biggest target. When does this go wrong?

Given 1 1 1, the greedy algorithm is indifferent to which cell it bombs first. Of course, the centre cell is better - it zeros all three cells at once. This suggests a new algorithm A, "bomb to minimise the sum remaining". When does this algorithm go wrong?

Given 1 1 2 1 1, algorithm A is indifferent between bombing the 2nd, 3rd or 4th cells. But bombing the 2nd cell to leave 0 0 1 1 1 is better than bombing the 3rd cell to leave 1 0 1 0 1. How to fix that? The problem with bombing the 3rd cell is that it leaves us work to the left and work to the right which must be done separately.

How about "bomb to minimise the sum remaining, but maximise the minimum to the left (of where we bombed) plus the minimum to the right". Call this algorithm B. When does this algorithm go wrong?

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Edit: After reading the comments, I agree a much more interesting problem would be the one dimensional problem changed so that the ends join up. Would love to see any progress on that.