# The distributions of functions related to parametric integer optimization

## TL;DR

This paper investigates the typical asymptotic behavior of functions related to integer programming, revealing that their usual values are smaller than worst-case bounds through a new probabilistic framework.

## Contribution

It introduces a framework for analyzing the asymptotic distribution of functions in integer optimization, focusing on typical rather than worst-case values.

## Key findings

- Typical values are smaller than worst-case bounds.
- Provides probabilistic results on the distribution of these functions.
- Framework applicable to general functions in integer optimization.

## Abstract

We consider the asymptotic distribution of the IP sparsity function, which measures the minimal support of optimal IP solutions, and the IP to LP distance function, which measures the distance between optimal IP and LP solutions. We create a framework for studying the asymptotic distribution of general functions related to integer optimization. There has been a significant amount of research focused around the extreme values that these functions can attain, however less is known about their typical values. Each of these functions is defined for a fixed constraint matrix and objective vector while the right hand sides are treated as input. We show that the typical values of these functions are smaller than the known worst case bounds by providing a spectrum of probability-like results that govern their overall asymptotic distributions.