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 about it so I changed them hoping to get a different perspective!).

• There are S stone throwers (slingers) and B birds.
• The slingers are not within the range of each other.
• Slinging once can kill all birds within the sight of a slinger and will consume one shot and one time unit

I am trying to figure out the optimal solution that minimizes the time and the number of shots it takes to kill the birds given a particular arrival pattern of birds. Let me give an example with absolute numbers: 3 Slingers and 4 birds.

        Time        1            2            3           4             5
Slinger
S1                B1, B2     B1, B2, B3       B4
S2                               B1         B1, B2      B3,B4     
S3                  B1         B3, B4                 B1,B2,B3,B4

and my data looks like this:

>> print t
[
  {
    1: {S1: [B1, B2], S2: [], S3: [B1]}, 
    2: {S1: [B1, B2, B3], S2: [B1], S3: [B3, B4]},
    3: {S1: [B4], S2: [B1,B2], S3: []},
    4: {S1: [], S2: [B3, B4], S3: [B1, B2, B3, B4]}
  }
]

There are a number of solutions I could think of (Sx at t=k implies that slinger Sx takes a shot at time k):

1. S1 at t=1, S1 at t=2, S1 at t=3 <- Cost: 3 shots + 3 time units = 6
2. S1 at t=2, S1 at t=3 <- Cost: 2 shots + 3 time units = 5
3. S1 at t=1, S3 at t=2 <- Cost: 2 shots + 2 time units = 4
4. S3 at t=4 <- Cost: 1 shot + 4 time units = 5

To me it appears that solution 3 is the optimal one in this. Of course, I did this by hand (so there is a possibility I may have missed something) but I cannot think of a scalable way of doing this. Also, I am worried there are corner cases because the decision of one shooter might alter the decision of others but because I have the global view, may be it does not matter.

What is a fast and good way to solving this problem in python? I am having a hard time coming up with a good data structure to do this leave alone the algorithm to do it. I am thinking of using dynamic programming because this seems to involve state space exploration but am a bit confused on how to proceed. Any suggestions?

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.