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

Mathematica Integer Linear Programming using branch-and-bound

As it has already been mentioned, this problem can be solved using integer linear programming (which is NP-Hard). Mathematica already has ILP built in. "To solve an integer linear programming problem Mathematica first solves the equational constraints, reducing the problem to one containing inequality constraints only. Then it uses lattice reduction techniques to put the inequality system in a simpler form. Finally, it solves the simplified optimization problem using a branch-and-bound method." [see Constrained Optimization Tutorial in Mathematica.. ]

I've written the following code that utilizes ILP libraries of Mathematica. It is surprisingly fast.

solveMatrixBombProblem[problem_, r_, c_] := 
 Module[{}, 
  bombEffect[x_, y_, m_, n_] := 
   Table[If[(i == x || i == x - 1 || i == x + 1) && (j == y || 
        j == y - 1 || j == y + 1), 1, 0], {i, 1, m}, {j, 1, n}];
  bombMatrix[m_, n_] := 
   Transpose[
    Table[Table[
      Part[bombEffect[(i - Mod[i, n])/n + 1, Mod[i, n] + 1, m, 
        n], (j - Mod[j, n])/n + 1, Mod[j, n] + 1], {j, 0, 
       m*n - 1}], {i, 0, m*n - 1}]];
  X := x /@ Range[c*r];
  sol = Minimize[{Total[X], 
     And @@ Thread[bombMatrix[r, c].X >= problem] && 
      And @@ Thread[X >= 0] && Total[X] <= 10^100 && 
      Element[X, Integers]}, X];
  Print["Minimum required bombs = ", sol[[1]]];
  Print["A possible solution = ", 
   MatrixForm[
    Table[x[c*i + j + 1] /. sol[[2]], {i, 0, r - 1}, {j, 0, 
      c - 1}]]];]

For the example provided in the problem:

solveMatrixBombProblem[{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}, 7, 5]

Outputs

For anyone reading this with a greedy algorithm

Try your code on the following 10x10 problem:

5   20  7   1   9   8   19  16  11  3  
17  8   15  17  12  4   5   16  8   18  
4   19  12  11  9   7   4   15  14  6  
17  20  4   9   19  8   17  2   10  8  
3   9   10  13  8   9   12  12  6   18  
16  16  2   10  7   12  17  11  4   15  
11  1   15  1   5   11  3   12  8   3  
7   11  16  19  17  11  20  2   5   19  
5   18  2   17  7   14  19  11  1   6  
13  20  8   4   15  10  19  5   11  12

Here it is comma-seperated:

5, 20, 7, 1, 9, 8, 19, 16, 11, 3, 17, 8, 15, 17, 12, 4, 5, 16, 8, 18, 4, 19, 12, 11, 9, 7, 4, 15, 14, 6, 17, 20, 4, 9, 19, 8, 17, 2, 10, 8, 3, 9, 10, 13, 8, 9, 12, 12, 6, 18, 16, 16, 2, 10, 7, 12, 17, 11, 4, 15, 11, 1, 15, 1, 5, 11, 3, 12, 8, 3, 7, 11, 16, 19, 17, 11, 20, 2, 5, 19, 5, 18, 2, 17, 7, 14, 19, 11, 1, 6, 13, 20, 8, 4, 15, 10, 19, 5, 11, 12

For this problem, my solution contains 208 bombs. Here's a possible solution (I was able to solve this in about 12 seconds).

As a way to test the results Mathematica is producing, see if your greedy algorithm can do any better.