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
(Schrijver 1984, p. 318). 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 (Lancini and Pisanu 2026, p. 10).
Barbato, M. and Lancini, E. "The Stable Marriage Polytope Is Box-Totally Dual Integral." 8 Oct 2026. https://arxiv.org/abs/2610.12720.Lancini,
E. and Pisanu, F. "A Horizon Tour of Box-Total Dual Integrality." Comput.
Sci. Rev.61, 100928, 2026. https://doi.org/10.1016/j.cosrev.2026.100928.Schrijver,
A. "Total Dual Integrality from Directed Graphs, Crossing Families, and Sub-
and Supermodular Functions." In Progress in Combinatorial Optimization
(Ed. W. R. Pulleyblank). Toronto, Canada: Academic Press, pp. 315-361,
1984. https://ir.cwi.nl/pub/10159/10159D.pdf.
retains total
dual integrality after adding arbitrary rational coordinate bounds 
. Any lower or upper bound may be omitted
(Schrijver 1984, p. 318). The word "box" refers to these coordinatewise
restrictions.
