An Axiomatic Duality Framework for the Theta Body and Related Convex Corners

TL;DR

This paper develops an axiomatic duality framework for the Lovasz theta function and theta body, unifying various relaxations and introducing new concepts like Schur Lifting within convex optimization and graph theory.

Contribution

It introduces a minimal axiomatic framework for theta bodies, generalizes key properties, and proposes the novel concept of Schur Lifting, expanding the theoretical understanding of graph relaxations.

Findings

Unified axiomatic characterization of theta functions and bodies

Introduction of Schur Lifting as a dual to PSD Lifting

Generalized copositive formulation for fractional chromatic number

Abstract

Lovasz theta function and the related theta body of graphs have been in the center of the intersection of four research areas: combinatorial optimization, graph theory, information theory, and semidefinite optimization. In this paper, utilizing a modern convex optimization viewpoint, we provide a set of minimal conditions (axioms) under which certain key, desired properties are generalized, including the main equivalent characterizations of the theta function, the theta body of graphs, and the corresponding antiblocking duality relations. Our framework describes several semidefinite and polyhedral relaxations of the stable set polytope of a graph as generalized theta bodies. As a by-product of our approach, we introduce the notion of "Schur Lifting" of cones which is dual to PSD Lifting (more commonly used in SDP relaxations of combinatorial optimization problems) in our axiomatic generalization. We also generalize the notion of complements of graphs to diagonally scaling-invariant polyhedral cones. Finally, we provide a weighted generalization of the copositive formulation of the fractional chromatic number by Dukanovic and Rendl.