Minimum numbers of attacks needed [closed]

Answer 1

This reminds me the Set Cover Problem:

You have a set U that stores the N monsters: U = {1, 2, ..., N}, a set S of possible attacks. Each attack is as well a set of the indexes of the monster that the attack kills. In your example, S is S = { {1}, {2}, {}, {1}, {2} }. You have to find the smallest set C in S whose union is U.

Answer 2

This can be modelled as a graph problem.

Create a graph node for each row and column where there's a monster. Connect the nodes if a monster is on that row and that column.

This is a bipartite graph, and you want to do minimum vertex cover. König's theorem shows that for bipartite graphs the problem is equivalnt with the maximum matching problem which can be solved in polinomial time:

http://en.wikipedia.org/wiki/Maximum_matching#Maximum_matchings_in_bipartite_graphs