Circuit Walks in Integral Polyhedra

TL;DR

This paper explores the properties of circuit walks in integral polyhedra, introducing a hierarchy based on their behavior and classifying classical polyhedra within this framework, with implications for combinatorial optimization.

Contribution

It introduces a new hierarchy for integral polyhedra based on circuit walk behaviors and classifies several classical families within this hierarchy.

Findings

Circuit walks only stop at integer points or vertices in certain polyhedra.

Characterizations of simple polytopes where all circuit walks are edge walks.

Generalization of simplices and parallelotopes through these polytopes.

Abstract

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.