We are given two sets of points $\{a_1, \ldots a_n\}$ and $\{b_1, \ldots, b_n\}$ in Euclidean space ( say $2$ dimensional). We are looking for a bijection $a_i \mapsto b_{\sigma(i)}$ so that the sum of distances $$\sum_{i} |a_i - b_{\sigma(i)}|$$ is minimal. Is there an efficient algorithm to find such a bijection ( for large $n$)?
If the points are on the line the bijection is the one preserving the order, so that is not hard.
Here is an approach that does not give a solution in general: Find the pair $(a_i, b_j)$ with the smallest distance, and pair these points. Continue with the rest of the points in the same way.