Box-total dual integrality, box-integrality, and equimodular matrices

Abstract

Box-totally dual integral (box-TDI) polyhedra are polyhedra described by systems which yield strong min-max relations. We characterize them in several ways, involving the notions of principal box-integer polyhedra and equimodular matrices. A polyhedron is box-integer if its intersection with any integer box $$\{\ell \le x \le u\}$$ is integer. We define principally box-integer polyhedra to be the polyhedra P such that $$ kP $$ is box-integer whenever $$ kP $$ is integer. A rational $$r\times n$$ matrix is equimodular if it has full row rank and its nonzero $$r\times r$$ 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. Our main result is that the following statements are equivalent. Along our proof, we show that a polyhedral cone is box-TDI if and only if it is box-integer, and that these properties are carried over to its polar. We illustrate these charaterizations 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.