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.
Addendum:
as the counter example of Brendan McKay shows, swapping pairs of rows or columns may not suffice to find the optimal assignment.
A preprocessing step that can deal with the counter example would be to first rotate the columns until the diagonal-elements have optimal cost-sum and only then strive for further improvements via swapping pairs of rows or columns.
It may even be necessary to interleave line-swapping with column-rotations frequently so that the elements on the principal diagonal always have better cost-sum than the elements on other diagonals.
