# Phase transitions in integer linear problems

## TL;DR

This paper explores phase transitions in the solvability of sparse integer linear problems, revealing a Van-Der-Waals phase diagram and increased computational difficulty near critical points.

## Contribution

It introduces a phase diagram framework for understanding the computational complexity of integer linear problems based on ensemble analysis.

## Key findings

- Feasible solution space exhibits a Van-Der-Waals phase diagram.
- Computational difficulty peaks near the critical point.
- Increased complexity observed in the coexistence region.

## Abstract

The resolution of linear system with positive integer variables is a basic yet difficult computational problem with many applications. We consider sparse uncorrelated random systems parametrised by the density $c$ and the ratio $\alpha=N/M$ between number of variables $N$ and number of constraints $M$. By means of ensemble calculations we show that the space of feasible solutions endows a Van-Der-Waals phase diagram in the plane ($c$, $\alpha$). We give numerical evidence that the associated computational problems become more difficult across the critical point and in particular in the coexistence region.