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 ${\displaystyle Ax\leq b}$, where ${\displaystyle A}$ and ${\displaystyle b}$ are rational, is called totally dual integral (TDI) if for any ${\displaystyle c\in \mathbb {Z} ^{n}}$ such that there is a feasible, bounded solution to the linear program

${\displaystyle {\begin{aligned}&&\max c^{\mathrm {T} }x\\&&Ax\leq b,\end{aligned}}}$

there is an integer optimal dual solution.^[1][2][3]

Edmonds and Giles^[2] showed that if a polyhedron ${\displaystyle P}$ is the solution set of a TDI system ${\displaystyle Ax\leq b}$, where ${\displaystyle b}$ has all integer entries, then every vertex of ${\displaystyle P}$ 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^[1] showed that if ${\displaystyle P}$ is a polytope whose vertices are all integer valued, then ${\displaystyle P}$ is the solution set of some TDI system ${\displaystyle Ax\leq b}$, where ${\displaystyle b}$ is integer valued.

Note that TDI is a weaker sufficient condition for integrality than total unimodularity.^[4]

References

1. Giles, F.R.; W.R. Pulleyblank (1979). "Total Dual Integrality and Integer Polyhedra". Linear Algebra and its Applications. 25: 191–196. doi:10.1016/0024-3795(79)90018-1.
2. Edmonds, J.; R. Giles (1977). "A min-max relation for submodular functions on graphs". Annals of Discrete Mathematics. 1: 185–204.
3. Schrijver, A. (1981). "On Total Dual Integrality". Linear Algebra and its Applications. 38: 27–32. doi:10.1016/0024-3795(81)90005-7.
4. Chekuri, C. "Combinatorial Optimization Lecture Notes" (PDF).