# Minimal inequalities for an infinite relaxation of integer programs

## TL;DR

This paper proves that maximal S-free convex sets are polyhedra when S is the set of integral points in a rational polyhedron, extending Lovász's theorem and linking these sets to minimal inequalities in integer programming.

## Contribution

It extends Lovász's theorem to a broader class of sets and establishes a correspondence between maximal S-free convex sets and minimal inequalities.

## Key findings

- Maximal S-free convex sets are polyhedra for S as integral points in rational polyhedra.
- Established a one-to-one correspondence between maximal S-free convex sets and minimal inequalities.
- Extended classical results to new settings in integer programming.

## Abstract

We show that maximal $S$-free convex sets are polyhedra when $S$ is the set of integral points in some rational polyhedron of $\mathbb{R}^n$. This result extends a theorem of Lov\'asz characterizing maximal lattice-free convex sets. Our theorem has implications in integer programming. In particular, we show that maximal $S$-free convex sets are in one-to-one correspondence with minimal inequalities.