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

ファイルを差し替え（2021/10/21）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

Greedy systems of linear inequalities and lexicographically optimal solutions

The present note reveals the role of the concept of greedy system of linear inequalities played in connection with lexicographically optimal solutions on convex polyhedra and discrete convexity. The lexicographically optimal solutions on convex polyhedra represented by a greedy system of linear inequalities can be obtained by a greedy procedure, a special form of which is the greedy algorithm of J. Edmonds for polymatroids. We also examine when the lexicographically optimal solutions become integral. By means of the Fourier–Motzkin elimination Murota and Tamura have recently shown the existence of integral points in a polyhedron arising as a subdifferential of an integer-valued, integrally convex function due to Favati and Tardella [Murota and Tamura, Integrality of subgradients and biconjugates of integrally convex functions. Preprint arXiv:1806.00992v1 (2018)], which can be explained by our present result. A characterization of integrally convex functions is also given