I have a problem which I like and I love to think about solutions, but I’m stuck unfortunately. I hope you like it too. The problem states:
I have two lists of 2D points(say A and B) and need to pair up points from A with points from B, under the condition that the sum of the distances in all pairs is minimal. A pair contains one point from A and one from B, a point can be used only once, and as many as possible pairs should be created(i.e. min(length(A), length(B))).
I’ve made a simple example, where color denotes which list the point is from, and the black connections are the solution.
Although this is a nice problem and I suspect is NP-hard, it gets nicer. I can build on existing solutions. Suppose I have two lists and the corresponding solution(i.e. the set of pairs), then the problem I need to solve is to reoptimalize that solution when a point is added to or removed from either list.
I’ve unfortunately not been able to come up with any non-brute force algorithm yielding the optimal solution. I hope you can. Any algorithm is appreciated in any (pseudo) language, preferably C#.
This problem is solvable in polynomial time via the Hungarian algorithm. To get a square matrix, add dummy entries to the shorter list at “distance 0” from everything.