Small Chvatal rank

TL;DR

This paper introduces the small Chvatal rank (SCR), a new measure for the complexity of generating facet normals of integer hulls, and explores its properties, especially in the context of the stable set problem.

Contribution

It defines SCR, characterizes matrices with zero SCR via supernormality, and analyzes the appearance of facet normals in the stable set polytope within a limited number of procedure rounds.

Findings

Many stable set polytope facet normals appear within two rounds of SCR.

Supernormality generalizes unimodularity and characterizes matrices with zero SCR.

Lower bounds for SCR are established for general polytopes and those in the unit cube.

Abstract

We propose a variant of the Chvatal-Gomory procedure that will produce a sufficient set of facet normals for the integer hulls of all polyhedra {xx : Ax <= b} as b varies. The number of steps needed is called the small Chvatal rank (SCR) of A. We characterize matrices for which SCR is zero via the notion of supernormality which generalizes unimodularity. SCR is studied in the context of the stable set problem in a graph, and we show that many of the well-known facet normals of the stable set polytope appear in at most two rounds of our procedure. Our results reveal a uniform hypercyclic structure behind the normals of many complicated facet inequalities in the literature for the stable set polytope. Lower bounds for SCR are derived both in general and for polytopes in the unit cube.