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 $$c \in \mathbb{Z}^n$$ such that there is a feasible, bounded solution to the linear program

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