Handelman's hierarchy for the maximum stable set problem

TL;DR

This paper explores Handelman's hierarchy of linear programming relaxations for the maximum stable set problem, establishing bounds and exact values for specific graph classes and relating it to other hierarchies.

Contribution

It introduces bounds and exact values for Handelman's hierarchy rank in relation to graph properties, connecting it with other relaxation hierarchies.

Findings

Bounds on Handelman rank for perfect graphs.

Exact Handelman rank for vertex-transitive graphs.

Computed ranks for odd cycles, wheels, and their complements.

Abstract

The maximum stable set problem is a well-known NP-hard problem in combinatorial optimization, which can be formulated as the maximization of a quadratic square-free polynomial over the (Boolean) hypercube. We investigate a hierarchy of linear programming relaxations for this problem, based on a result of Handelman showing that a positive polynomial over a polytope with non-empty interior can be represented as conic combination of products of the linear constraints defining the polytope. We relate the rank of Handelman's hierarchy with structural properties of graphs. In particular we show a relation to fractional clique covers which we use to upper bound the Handelman rank for perfect graphs and determine its exact value in the vertex-transitive case. Moreover we show two upper bounds on the Handelman rank in terms of the (fractional) stability number of the graph and compute the Handelman rank for several classes of graphs including odd cycles and wheels and their complements. We also point out links to several other linear and semidefinite programming hierarchies.