 Solution

The assignment equalities parameterize every feasible matrix of the linear programming relaxation asThe side totals and objective becomeThe first side constraint gives $t\le 1/2$ and the second gives $t\ge 1/2$, so the relaxation has the unique feasible pointDirect elimination of the assignment equalities has solved this one-dimensional linear program. Its value matches the Lagrangian dual problem value because the retained assignment polytope is the convex hull of the two integer assignments; it does not make the side-constrained integer problem feasible.