Box-Total Dual Integrality, Box-Integrality, and Equimodular Matrices

A polyhedron is box-integer if its intersection with any integer box {ℓ≤x≤u} is integer. We define principally box-integer polyhedra to be the polyhedra P such that kP is box-integer whenever kP is integer. We characterize them in several ways, involving equimodular matrices and box-total dual integral (box-TDI) systems. A rational r×n matrix is equimodular if it has full row rank and its nonzero r×r determinants all have the same absolute value. A face-defining matrix is a full row rank matrix describing the affine hull of a face of the polyhedron. Box-TDI systems are systems which yield strong min-max relations, and the underlying polyhedron is called a box-TDI polyhedron. Our main result is that the following statements are equivalent.
- The polyhedron P is principally box-integer.
- The polyhedron P is box-TDI.
- Every face-defining matrix of P is equimodular.
- Every face of P has an equimodular face-defining matrix.
- Every face of P has a totally unimodular face-defining matrix.
- For every face F of P, lin(F) has a totally unimodular basis.
Along our proof, we show that a cone {x:Ax≤0} is box-TDI if and only if it is box-integer, and that these properties are passed on to its polar.
We illustrate the use of these characterizations by reviewing well known results about box-TDI polyhedra. We also provide several applications. The first one is a new perspective on the equivalence between two results about binary clutters. Secondly, we refute a conjecture of Ding, Zang, and Zhao about box-perfect graphs. Thirdly, we discuss connections with an abstract class of polyhedra having the Integer Carathéodory Property. Finally, we characterize the box-TDIness of the cone of conservative functions of a graph and provide a corresponding box-TDI system.