On integer program with bounded minors and flatness theorem

Let A be an m × n integral matrix of 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, where k is a some natural constant. We will call that matrices like k-modular. We investigate an integer program max{cx : Ax ≤ b, x ∈ Zn} where A is k-modular. We say that A is almost unimodular (see [2, 5]) if it is 2-modular and the absolute values of the (n− 1)× (n− 1) minors is at most 1. We also call 2-modular matrices like bimodular (see [1]). Let γ(P, c) = max{cx : x ∈ P}−min{cx : x ∈ P}. We use the notion of the width of a polyhedron (see [6]): width(P ) = min{γ(P, c) : c ∈ Zn \ {0}} The presentation consists of several sections: • In the first section, we study the problem of integer programming, some polynomial cases and recent results about the integer program with bounded minors. • In the second section we show that the integer program with the almost unimodular matrix can be solved in polynomial time, based on recent work of Chirkov and Veselov. • In the third section, we consider the concept of width of polyhedron and study some important results about the minimal width and the flatness of polyhedron. Also we study, how to use the width of polyhedron in a practice. • Finally, we consider recent results, that combines results about integer programs with bounded minors and the notion of minimal width of polyhedron.