Finding an optimal solution that minimizes a constraint?

Question

Let us call this problem the Slinger-Bird problem (actually the Slinger is analogous to a server and the bird to a request but I was having a nervous breakdown thinking abou

Answer 1

This is not an optimal assignment problem, because slingers kill all birds in view.

You have a two-dimensional objective function, so there can be a number of tradeoffs between shots and time. Determining the minimum number of shots for a particular time limit is exactly the set cover problem (as mhum suggests). The set cover problem is NP-hard and hard to approximate, but in practice, branch and bound using the dual of the linear programming formulation is quite effective in finding the optimum.