Circuits play a fundamental role in the theory of linear programming due to their intimate connection to algorithms of combinatorial optimization and the efficiency of the simplex method. We are interested in better understanding the properties of circuit walks in integral polyhedra. In this paper, we introduce a hierarchy for integral polyhedra based on different types of behavior exhibited by their circuit walks. Many problems in combinatorial optimization fall into the most interesting categories of this hierarchy — steps of circuit walks only stop at integer points, at vertices, or follow actual edges. We classify several classical families of polyhedra within the hierarchy, including $0∕1$-polytopes, polyhedra defined by totally unimodular matrices, and more specifically matroid polytopes, transportation polytopes, and partition polytopes. Finally, we prove three characterizations of the simple polytopes that appear in the bottom level of the hierarchy where all circuit walks are edge walks, showing that such polytopes constitute a generalization of simplices and parallelotopes.

电路在线性规划理论中起着基本作用，因为它们与组合优化算法密切相关，并且单纯形法的效率很高。我们有兴趣更好地了解整体多面体中电路行走的特性。在本文中，我们根据电路游走所表现出的不同类型的行为，为积分多面体引入了层次结构。组合优化中的许多问题都属于该层次结构中最有趣的类别-电路行走的步骤仅停止在整数点，顶点或沿实际边沿。我们在层次结构中将几个经典的多面体家族分类，包括$0∕\mathrm{1个}$-多面体，由完全单模矩阵定义的多面体，更具体地说是拟阵多面体，运输多面体和分隔多面体。最后，我们证明了出现在所有电路走线都是边缘走线的层次结构最底层的简单多面体的三个特征，表明这种多面体构成了单纯形体和平行体的概括。