A Note on Integrality of Convex Polyhedra Represented by Linear Inequalities with {0,±1}-coefficients

Abstract

We consider a polyhedron P represented by linear inequalities with {0, ±1}-coefficients. We show a condition that guarantees existence of an integral vector in P, which also turns out to be an extreme point of P. We reveal how our polyhedral and geometric approach shows the recent interesting integrality results of Murota and Tamura about subdifferentials of integrally convex functions. Their proofs are algebraic, based on the Fourier-Motzkin elimination for the relevant systems of linear inequalities. Our approach provides further insight into subdifferentials of integrally convex functions to fully appreciate the integrality results of Murota and Tamura from a polyhedral and geometric point of view.