Total dual integrality and integer polyhedra

Abstract

A linear system Ax ⩽ b ( A, b rational) is said to be totally dual integral (TDI) if for any integer objective function c such that max { cx: Ax ⩽ b} exists, there is an integer optimum dual solution. We show that if P is a polytope all of whose vertices are integer valued, then it is the solution set of a TDI system Ax ⩽ b where b is integer valued. This was shown by Edmonds and Giles [4] to be a sufficient condition for a polytope to have integer vertices.