Pair two sets such that the distance between elements is minimized

Question

I have two sets S_1 and S_2. Given these two sets, I need to pair each element from S_1 with an element from S_2.

• The elements are not reusable, so if S_1[A] is paired with S_2[D], then I cannot also pair S_1[B] with S_2[D].
• The goal is to produce a pairing using all elements such that the distance of the pairing is minimized.
• The distance of the pairing is computed as the sum of the distance between each pair.
• Produce result with lowest total paired points value

Are there any known algorithms for solving this type of problem efficiently?

Part of the difficulty is that taking a greedy approach doesn't work. If S_1 = [A, B, C] and S_2 = [D, E, F], and distance(A, D) = 0.1, distance(A, E) = 0.3, distance(A, F) = 0.4, you can't naively match A to D just because it has the lowest distance for this set. Suppose that distance(B, D) = 0.1, distance(B, E) = 0.8, and distance(B, F) = 0.9. If you naively choose to match (A, D) in the first iteration, then you actually make the overall distance higher because this forces you to match either (B, E) or (B, D). It would be a better choice to match (A, E) and then allow (B, D) to match. This means you can't iterate over S_1 and greedily assign matches based on the lowest distance between each element of S_1 and the remaining elements of S_2.

This seems similar to the assignment problem, which I could solve using something like the Hungarian Algorithm (https://en.wikipedia.org/wiki/Hungarian_algorithm), but I believe that algorithm allows reusing elements, which won't work for my case.