On integer programing with bounded determinants

TL;DR

This paper investigates conditions under which integer points exist within polytopes defined by matrices with bounded determinants, providing new theoretical results and polynomial-time algorithms for specific classes of matrices.

Contribution

It establishes new criteria for the existence of integer points in polytopes based on matrix determinants and width, and introduces polynomial algorithms for certain matrix classes.

Findings

If all r(A) x r(A) sub-matrices have determinants ±Δ(A) or 0 and width is sufficiently large, then the polytope contains n affine independent integer points.

For simplices with width ≥ δ(A)-1, a polynomial-time algorithm finds an integer point.

Integer programming is polynomial-time solvable when A is almost unimodular.

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 $\Delta(A)$ and $\delta(A)$ 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 $\pm \Delta(A)$ or $0$ and $w(P)\ge (\Delta(A)-1)(n+1)$, 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 $w(P)\ge \delta(A)-1$, 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 $\Delta(A) \leq 2$ and any $(r(A)-1)\times(r(A)-1)$ sub-matrix has a determinant from the set $\{0,\pm 1\}$.