Every rational polyhedron has finite split rank: new proof

TL;DR

This paper presents a new proof demonstrating that the split rank of any rational polyhedron is finite, using a potential function approach rather than traditional methods based on Chvátal closures.

Contribution

It introduces an independent proof for finite split rank of rational polyhedra, employing a potential function decreasing with each split closure step.

Findings

Split rank of rational polyhedra is finite.

A potential function decreases with each split closure.

Provides an alternative proof method for a known result.

Abstract

Split rank of a rational polyhedron is finite. The well known proof of this is based on the fact that split closure is stronger than the Chv\'{a}tal closure, and the Chv\'{a}tal rank of a rational polyhedron is finite due to the result of Chv\'{a}tal and Schrijver. In this note we provide an independent proof for the fact that every rational polyhedron has finite split rank. In principal, we construct a nonnegative potential function which decreases by at least one with "every" second split closure unless the integer hull of the polyhedron is reached.