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

For updated question a simple greedy algorithm gives optimal result.

Drop A[0,0] bombs to cell A[1,1], then drop A[1,0] bombs to cell A[2,1], and continue this process downwards. To clean bottom left corner, drop max(A[N-1,0], A[N-2,0], A[N-3,0]) bombs to the cell A[N-2,1]. This will completely clean up first 3 columns.

With the same approach clean columns 3,4,5, then columns 6,7,8, etc.

Unfortunately this does not help finding solution for the original problem.

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"Larger" problem (without "nonicreasing" constraint) may be proven to be NP-hard. Here is sketch of a proof.

Suppose we have a planar graph of degree up to 3. Let's find minimum vertex cover for this graph. According to Wikipedia article this problem is NP-hard for planar graphs of degree up to 3. This could be proven by reduction from Planar 3SAT. And hardness of Planar 3SAT - by reduction from 3SAT. Both these proofs are presented in recent lectures in "Algorithmic Lower Bounds" by prof. Erik Demaine (lectures 7 and 9).

If we split some edges of the original graph (left graph on the diagram), each one with even number of additional nodes, the resulting graph (right graph on the diagram) should have exactly the same minimum vertex cover for original vertices. Such transformation allows to align graph vertices to arbitrary positions on the grid.

If we place graph vertices only to even rows and columns (in such a way that no two edges incident to one vertex form an acute angle), insert "ones" wherever there is an edge, and insert "zeros" to other grid positions, we could use any solution for the original problem to find minimum vertex cover.