Total dual integrality

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Let [math]A \in {\mathbb Q}^{m\times n}[/math] be a rational matrix, and let [math]b \in {\mathbb Q}^m[/math]. The linear inequality system [math]Ax \leq b[/math] is totally dual integral (TDI for short) if for every integer vector [math]c \in {\mathbb Z}^n[/math] the system [math]\max\{yb: yA=c,\ y \geq 0\}[/math] 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 [math]Ax \leq b[/math] is TDI and the vector b is integer, then [math]\{x: Ax \leq b\}[/math] is an integer polyhedron.

Another definition can be given using the notion of Hilbert basis. A system [math]Ax \leq b[/math] is TDI if and only if for every face F of the polyhedron [math]P=\{x: Ax \leq b\}[/math], 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 [math]Ax \leq b[/math] such that the matrix A is integral. If P is an integer polyhedron, then b can also be integral.


The following are some examples of TDI systems:


  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