On integer programming with bounded determinants

Let A be an $$(m \times n)$$(m×n) integral matrix, and let $$P=\{ x :A x \le b\}$$P={x:Ax≤b} be an n-dimensional polytope. The width of P is defined as $$ w(P)=min\{ x\in \mathbb {Z}^n{\setminus }\{0\} :max_{x \in P} x^\top u - min_{x \in P} x^\top v \}$$w(P)=min{x∈Zn\{0}:maxx∈Px⊤u-minx∈Px⊤v}. Let $$\varDelta (A)$$Δ(A) and $$\delta (A)$$δ(A) denote the greatest and the smallest absolute values of a determinant among all $$r(A) \times r(A)$$r(A)×r(A) sub-matrices of A, where r(A) is the rank of the matrix A. We prove that if every $$r(A) \times r(A)$$r(A)×r(A) sub-matrix of A has a determinant equal to $$\pm \varDelta (A)$$±Δ(A) or 0 and $$w(P)\ge (\varDelta (A)-1)(n+1)$$w(P)≥(Δ(A)-1)(n+1), then P contains n affine independent integer points. Additionally, we present similar results for the case of k-modular matrices. The matrix A is called totallyk-modular if every square sub-matrix of A has a determinant in the set $$\{0,\, \pm k^r :r \in \mathbb {N} \}$${0,±kr:r∈N}. When P is a simplex and $$w(P)\ge \delta (A)-1$$w(P)≥δ(A)-1, we describe a polynomial time algorithm for finding an integer point in P.