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.
Your problem is an instance of the weighted minimum maximal matching problem (as described in this Wikipedia article). There is no polynomial-time algorithm even for the unweighted problem (all distances equal). There are efficient algorithms to approximately solve it in polynomial time (within a factor of 2).
This is the minimum weight Euclidean bipartite matching problem. There is a O(n^(2+epsilon)) algorithm.
来源：https://stackoverflow.com/questions/6091567/optimization-from-partial-solution-minimize-sum-of-distances-between-pairs