Persistency of Linear Programming Relaxations for the Stable Set Problem

TL;DR

This paper investigates the persistency property of linear programming relaxations for the stable set problem, showing that only the standard LP formulation can have this property across all graphs.

Contribution

It proves that no alternative LP formulations satisfying mild conditions possess the persistency property unless they are equivalent to the stable set polytope.

Findings

Standard LP has persistency property for stable set problem

Other LP formulations lack persistency unless identical to the stable set polytope

Persistency is unique to the standard LP formulation across all graphs

Abstract

The Nemhauser-Trotter theorem states that the standard linear programming (LP) formulation for the stable set problem has a remarkable property, also known as (weak) persistency: for every optimal LP solution that assigns integer values to some variables, there exists an optimal integer solution in which these variables retain the same values. While the standard LP is defined by only non-negativity and edge constraints, a variety of other LP formulations have been studied and one may wonder whether any of them has this property as well. We show that any other formulation that satisfies mild conditions cannot have the persistency property on all graphs, unless it is always equal to the stable set polytope.