On Connections Between Association Schemes and Analyses of Polyhedral and Positive Semidefinite Lift-and-Project Relaxations

TL;DR

This paper investigates the relationship between association schemes and SDP-based relaxations in combinatorial optimization, deriving bounds and analyzing the structure of relaxations for problems like stable set and hypergraph matching.

Contribution

It establishes new connections between association schemes and lift-and-project relaxations, providing bounds and structural insights for combinatorial optimization problems.

Findings

Bounds on clique and stability numbers derived from association schemes

Exact or bounded lift-and-project ranks for hypergraph matching relaxations

Introduction of a non-commutative association scheme based on hypermatchings

Abstract

We explore some connections between association schemes and the analyses of the semidefinite programming (SDP) based convex relaxations of combinatorial optimization problems in the Lov\'{a}sz--Schrijver lift-and-project hierarchy. Our analysis of the relaxations of the stable set polytope leads to bounds on the clique and stability numbers of some regular graphs reminiscent of classical bounds by Delsarte and Hoffman, as well as the notion of deeply vertex-transitive graphs -- highly symmetric graphs that we show arise naturally from some association schemes. We also study relaxations of the hypergraph matching problem, and determine exactly or provide bounds on the lift-and-project ranks of these relaxations. Our proofs for these results also inspire the study of a homogeneous coherent configuration based on hypermatchings, which is an association scheme except it is generally non-commutative. We then illustrate the usefulness of obtaining commutative subschemes from non-commutative homogeneous coherent configurations via contraction in this context.