A Note on Non-Degenerate Integer Programs with Small Sub-Determinants

TL;DR

This paper presents a polynomial-time algorithm for solving certain non-degenerate integer programs with small sub-determinants, expanding the class of problems efficiently solvable in integer optimization.

Contribution

The paper introduces an algorithm that solves integer programs with bounded sub-determinants in polynomial time, under specific conditions on matrix entries and sub-determinants.

Findings

Polynomial-time solution for integer programs with bounded sub-determinants.

Applicable to problems where matrix entries and sub-determinants are small and constant.

Extends the class of efficiently solvable integer optimization problems.

Abstract

The intention of this note is two-fold. First, we study integer optimization problems in standard form defined by $A \in\mathbb{Z}^{m\times{}n}$ and present an algorithm to solve such problems in polynomial-time provided that both the largest absolute value of an entry in $A$ and $m$ are constant. Then, this is applied to solve integer programs in inequality form in polynomial-time, where the absolute values of all maximal sub-determinants of $A$ lie between $1$ and a constant.