On integer programing with restricted determinants

Let A be an (m × n) integral matrix, and let P = {x : Ax ≤ b} be an n-dimensional polytope. The width of P is defined as w(P ) = min{x ∈ Z\{0} : maxx∈Pxu−minx∈Pxv}. Let ∆(A) and δ(A) denote the greatest and the smallest absolute values of a determinant among all r(A)× r(A) submatrices of A, where r(A) is the rank of a matrix A. We prove that if every r(A) × r(A) sub-matrix of A has a determinant equal to ±∆(A) or 0 and w(P ) ≥ (∆(A) − 1)(n + 1), then P contains n affine independent integer points. Also we have similar results for the case of k-modular matrices. The matrix A is called totally k-modular if every square sub-matrix of A has a determinant in the set {0, ±k : r ∈ N}. When P is a simplex and w(P ) ≥ δ(A)− 1, we describe a polynomial time algorithm for finding an integer point in P . Finally we show that if A is almost unimodular, then integer program max{cx : x ∈ P ∩ Z} can be solved in polynomial time. The matrix A is called almost unimodular if ∆(A) ≤ 2 and any (r(A) − 1) × (r(A) − 1) sub-matrix has a determinant from the set {0,±1}.