Here's my attempt to explain the problem in mathematical language:

$$ \text{Given square matrix A} $$ $$ \left( \begin{array}{cccc} a_{1,1} & a_{1,2} & \cdots & a_{1,N} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,N} \\ \vdots & \vdots & \ddots & \vdots \\ a_{N,1} & a_{N,2} & \cdots & a_{N,N} \end{array} \right) $$

Find a minimal sum of $N$ elements such that no two elements are in the same row or column. In other words, find $$\min_{\sigma\in S_N}\left\{\sum_{i=1}^Na_{i,\sigma(i)}\right\},$$ where $S_N$ is the set of permutations on $\{1,\ldots,N\}$.

Is there a standard name for this problem?


This is called the (linear) assignment problem. It can be solved efficiently through linear programming over the Birkhoff polytope.

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