On integer programing with bounded determinants

Abstract

Let $A$ be an $(m \times n)$ integral matrix, and let $P=\{ x : A x \leq b\}$ 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 \}$. Let $Δ(A)$ and $δ(A)$ denote the greatest and the smallest absolute values of a determinant among all $r(A) \times r(A)$ sub-matrices of $A$, where $r(A)$ is the rank of a matrix $A$. We prove that if every $r(A) \times r(A)$ sub-matrix of $A$ has a determinant equal to $\pm Δ(A)$ or $0$ and $w(P)\ge (Δ(A)-1)(n+1)$, then $P$ contains $n$ affine independent integer points. Also we have similar results for the case of \emph{$k$-modular} matrices. The matrix $A$ is called \emph{totally $k$-modular} if every square sub-matrix of $A$ has a determinant in the set $\{0,\, \pm k^r :\: r \in \mathbb{N} \}$. When $P$ is a simplex and $w(P)\ge δ(A)-1$, we describe a polynomial time algorithm for finding an integer point in $P$. Finally we show that if $A$ is \emph{almost unimodular}, then integer program $\max \{c^\top x :\: x \in P \cap \mathbb{Z}^n \}$ can be solved in polynomial time. The matrix $A$ is called \emph{almost unimodular} if $Δ(A) \leq 2$ and any $(r(A)-1)\times(r(A)-1)$ sub-matrix has a determinant from the set $\{0,\pm 1\}$.