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, where and are rational, is called totally dual integral if for any such that there is a feasible, bounded solution to the linear program
there is an integer optimal dual solution.
Edmonds and Giles showed that if a polyhedron is the solution set of a TDI system, where has all integer entries, then every vertex of 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 is a polytope whose vertices are all integer valued, then is the solution set of some TDI system, where is integer valued.
Note that TDI is a weaker sufficient condition for integrality than total unimodularity.
