Optimization from partial solution: minimize sum of distances between pairs

Question

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 point

Answer 1

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.

Answer 2

This is the minimum weight Euclidean bipartite matching problem. There is a O(n^(2+epsilon)) algorithm.

Answer 3

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).