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I came across a problem which I think can be reduced to the assignment problem/Hungarian algorithm.

We have matrix $A$ and matrix $B$ which are both $n\times n$ symmetric matrices. We can rearrange $B$ in the following manner: column $i$ can be swapped with column $j$ provided that next, row $i$ is swapped with row $j$. This preserves the symmetry of $B$.

The problem statement is to find a rearrange of $B$ in this manner which minimizes the Frobenius norm of $A-B$.

Any guidance on an approach would be greatly appreciated. Thank you very much in advance.

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