# Interpreting optimal matchings as permutations

If $$\boldsymbol{A}\in\mathbb{R}^{n\times n}$$ is the cost-matrix of an assignment problem, then the usual statement of the problem of finding an optimal assignment is to identify $$n$$ elements $$a_{i,\,\pi(i)},\ i=1\cdots n$$ of least cost-sum, i.e. to directly determine the solution set from $$\boldsymbol{A}$$ by modifying its entries e.g. according to the Hungarian algorithm.

Question:

can the following interpretation of the assignment problem fail to report the optimal solution:

determine the sequence of line-exchanges that renders the sum of the diagonal-elements optimal?

If the permutation-formulation also generates the optimal solution to the assignment problem, that would yield a greedy algorithm:

Exchanging two lines also exchanges two on-diagonal elements with two off-diagonal elements with known effect on the cost-sum of the elements on the diagonal.
It doesn't seem reasonable to exchange a pair of lines that doesn't bring about the maximal cost-reduction for the elements on the diagonal.

$$\pmatrix{ 2&3&0&0\\0&2&3&0\\0&0&2&3\\3&0&0&2}$$