# Total dual integrality

Let $A \in {\mathbb Q}^{m\times n}$ be a rational matrix, and let $b \in {\mathbb Q}^m$. The linear inequality system $Ax \leq b$ is totally dual integral (TDI for short) if for every integer vector $c \in {\mathbb Z}^n$ the system $\max\{yb: yA=c,\ y \geq 0\}$ has an integer optimal solution provided that the optimum is finite.

This notion was introduced by Edmonds and Giles [1], who proved the following fundamental property of TDI systems.

Theorem. If the system $Ax \leq b$ is TDI and the vector b is integer, then $\{x: Ax \leq b\}$ is an integer polyhedron.

Another definition can be given using the notion of Hilbert basis. A system $Ax \leq b$ is TDI if and only if for every face F of the polyhedron $P=\{x: Ax \leq b\}$, the rows of A that correspond to tight inequalities for F form a Hilbert basis. This characterization can be used to prove the following theorem of Giles and Pulleyblank [2]:

Theorem. If P is a rational polyhedron, then it has a TDI linear description $Ax \leq b$ such that the matrix A is integral. If P is an integer polyhedron, then b can also be integral.

## Examples

The following are some examples of TDI systems:

## References

1. J. Edmonds, R. Giles, A min-max relation for submodular functions on graphs, in: Studies in Integer Programming (Proceedings of the Workshop on Integer Programming, Bonn, 1975; P.L. Hammer, E.L. Johnson, B.H. Korte, G.L. Nemhauser, eds.), 1977, 185–204. DOI link, Google Books link.
2. F.R. Giles, W.R. Pulleyblank, Total dual integrality and integer polyhedra, DOI link