# The Integrality Number of an Integer Program

## TL;DR

This paper introduces the concept of the integrality number for integer programs, showing how it relates to the structure of the constraint matrix and enabling solutions with fewer integer constraints.

## Contribution

It defines the integrality number of an IP, analyzes its invariance under unimodular transformations, and establishes bounds relating the number of constraints to the matrix's largest minor.

## Key findings

- IPs with n constraints can be solved with O(√Δ) integer constraints
- Integrality number is invariant under unimodular transformations
- Provides bounds linking the number of variables, constraints, and matrix minors

## Abstract

We introduce the integrality number of an integer program (IP) in inequality form. Roughly speaking, the integrality number is the smallest number of integer constraints needed to solve an IP via a mixed integer (MIP) relaxation. One notable property of this number is its invariance under unimodular transformations of the constraint matrix. Considering the largest minor $\Delta$ of the constraint matrix, our analysis allows us to make statements of the following form: there exist numbers $\tau(\Delta)$ and $\kappa(\Delta)$ such that an IP with $n\geq \tau(\Delta)$ many variables and $n + \kappa(\Delta)\cdot \sqrt{n}$ many inequality constraints can be solved via a MIP relaxation with fewer than $n$ integer constraints. From our results it follows that IPs defined by only $n$ constraints can be solved via a MIP relaxation with $O(\sqrt{\Delta})$ many integer constraints.