The Flatness Theorem for Some Class of Polytopes and Searching an Integer Point

Abstract

Let A be an m × n integral matrix of the rank n, we say that A has bounded minors if the maximum of the absolute values of the n × n minors is at most k, we will call these matrices as k-modular. We investigate an integer program $\mathit{max}\{c^{\top }x: Ax \leq b,x \in \mathbb{Z}^{n}\}$, where A is k-modular. We say that A is almost unimodular if it is 2-modular and the absolute values of its $(n - 1) \times (n - 1)$ minors are at most 1. We also refer 2-modular matrices to as bimodular. We say that A is strict k-modular if the absolute values of its n × n minors are from set {0, k, −k}. We prove that the width of an empty lattice polytope is less than $(k - 1)\,(n + 1)$ if it is induced by a system of inequalities with a strict k-modular matrix. Furthermore, we can give a polynomial-time algorithm for searching an integer point in a strict k-modular polytope if its width is grater than $(k - 1)\,(n + 1)$.