Total dual integrality

In mathematical optimization, total dual integrality is a sufficient condition for the integrality of a polyhedron. Thus, the optimization of a linear objective over the integral points of such a polyhedron can be done using techniques from linear programming. A linear system $Ax\le b$, where $A$ and $b$ are rational, is called totally dual integral (TDI) if for any $c\in {\mathbb{\mathbb{Z}}}^n$ such that there is a feasible, bounded solution to the linear program

$\begin{array}{rlr}& & \max c^{\mathrm{T}}x\\ & & Ax\le b,\end{array}$

there is an integer optimal dual solution. Edmonds and Giles showed that if a polyhedron $P$ is the solution set of a TDI system $Ax\le b$, where $b$ has all integer entries, then every vertex of $P$ is integer-valued. Thus, if a linear program as above is solved by the simplex algorithm, the optimal solution returned will be integer. Further, Giles and Pulleyblank showed that if $P$ is a polytope whose vertices are all integer valued, then $P$ is the solution set of some TDI system $Ax\le b$, where $b$ is integer valued. Note that TDI is a weaker sufficient condition for integrality than total unimodularity.