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performance tuning - How to get a pair with smallest total distance from one list in high-efficiency method


This is similar to this post, but actually more complicated. I have two lists in that question, and I want to get a pair. But in here, I want to get a pair from one list (even points). Given these 20 points:


SeedRandom[1]

pts = RandomInteger[20, {20, 2}]


{{5,0},{7,0},{2,3},{0,0},{16,14},{3,8},{19,5},{18,16},{12,0},{19,4},{7,3},{0,4},{20,3},{5,12},{19,8},{11,2},{3,10},{4,2},{17,11},{15,6}}



I can use FindIndependentEdgeSet to get one pair like this:


g = CompleteGraph[20];
List @@@ FindIndependentEdgeSet[VertexReplace[g, Thread[VertexList[g] -> pts]]]



{{{5,0},{7,0}},{{2,3},{3,8}},{{0,0},{16,14}},{{19,5},{4,2}},{{18,16},{3,10}},{{12,0},{7,3}},{{19,4},{11,2}},{{0,4},{20,3}},{{5,12},{19,8}},{{17,11},{15,6}}}



The total distance is


Total[EuclideanDistance @@@ pair] // N


113.859



But I'm sure it is not the smallest pair. Actually, I think I need a FindIndependentEdgeSet of edge weight version, but it seems the FindIndependentEdgeSet regards weighted graph as unweighted directly. Can anyone give me advice? Of course, I'm happy to know other methods that can do this which aren't based on Graph Theory.




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