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 $$\{\ell \le x \le u\}$$ { ℓ ≤ x ≤ u } is integer. We define principally box-integer polyhedra to be the polyhedra P such that $$ kP $$ kP is box-integer whenever $$ kP $$ kP is integer. A rational $$r\times n$$ r × n matrix is equimodular if it has full row rank and its nonzero $$r\times r$$ 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. Our main result is that the following statements are equivalent. The polyhedron P is box-TDI. The polyhedron P is principally box-integer. 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 , $$\mathrm{lin}(F)$$ lin ( F ) has a totally unimodular basis. 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.

中文翻译：

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盒全对偶积分、盒积分和等模矩阵

盒全对偶积分 (box-TDI) 多面体是由产生强最小-最大关系的系统描述的多面体。我们以多种方式描述它们，包括主盒整数多面体和等模矩阵的概念。如果多面体与任何整数框 $$\{\ell \le x \le u\}$$ { ℓ ≤ x ≤ u } 的交集是整数，则该多面体是框整数。我们主要将盒整数多面体定义为多面体 P，这样只要 $$ kP $$ kP 是整数，$$ kP $$ kP 就是盒整数。一个有理 $$r\times n$$ r × n 矩阵是等模的，如果它具有完整的行秩并且它的非零 $$r\times r$$ r × r 行列式都具有相同的绝对值。面定义矩阵是描述多面体面的仿射壳的全行秩矩阵。我们的主要结果是以下语句是等效的。多面体 P 是 box-TDI。多面体 P 主要是盒整数。P 的每个面定义矩阵都是等模的。P 的每个面都有一个等模面定义矩阵。P 的每个面都有一个完全单模的面定义矩阵。对于 P 的每个面 F，$$\mathrm{lin}(F)$$ lin ( F ) 都有一个完全单模的基。在我们的证明过程中，我们证明了多面体锥体是 box-TDI 当且仅当它是 box-integer，并且这些特性会延续到它的极坐标。我们通过回顾有关 box-TDI 多面体的众所周知的结果来说明这些特征。我们还提供多种应用程序。第一个是关于二元杂波的两个结果之间的等价性的新观点。其次，我们驳斥了丁、臧和赵关于完美盒图的猜想。第三，我们讨论与具有整数 Carathéodory 属性的抽象类多面体的联系。最后，我们表征了图的保守函数锥的 box-TDIness 并提供了相应的 box-TDI 系统。