Box-total dual integrality, abbreviated box-TDI, is the property that a rational system of linear inequalities 
retains total
dual integrality after adding arbitrary rational coordinate bounds 
. Any lower or upper bound may be omitted (Barbato
and Lancini 2026). The word "box" refers to these coordinatewise restrictions.
For every integer objective vector whose maximum over the augmented system is finite, the corresponding dual problem must have an integer optimal solution. The bounds themselves need not be integers. A rational polyhedron is box-totally dual integral if it admits a box-TDI defining system (Barbato and Lancini 2026).
Barbato and Lancini (2026) proved that the stable marriage polytope is box-totally dual integral, and that Rothblum's defining system has this property.
