Characterization of faceted near-TDI systems that are TDI

Characterize when a near-TDI faceted system is totally dual integral (TDI).

Background

The paper defines a system of integral linear inequalities Mx≤b to be near-TDI when it is totally dual in the p-adic rationals for every prime p. A faceted system is an irredundant, full-dimensional integral description whose individual constraints are primitive, meaning that the greatest common divisor of the coefficients on the left-hand side and the right-hand side is one.

The authors prove that non-degenerate near-TDI systems are TDI and characterize non-degenerate TDI systems through a Hilbert-cone condition together with resiliency. They explicitly leave unresolved the broader question of identifying exactly which near-TDI faceted systems are TDI, including potentially degenerate systems.