Total dual integrality is a property of a rational system of linear inequalities 
.
The system is totally dual integral if, for every integer
objective vector 
for which 
is finite, the dual problem
 |
has an integer optimal solution (Barbato and Lancini 2026). The property belongs to the defining system, not just its solution polyhedron.
Box-total dual integrality strengthens this property by requiring it to survive the addition of arbitrary rational coordinate
bounds 
.
Barbato and Lancini (2026) proved that the stable
marriage polytope admits such a defining system, and that Rothblum's defining
system itself has this stronger property.
