Theta Bodies for Polynomial Ideals

TL;DR

This paper introduces theta bodies, a hierarchy of semidefinite relaxations for polynomial ideals, unifying and extending existing relaxations like Lovász's theta body, with applications to combinatorial optimization problems.

Contribution

It establishes theta bodies as a canonical, computable hierarchy of relaxations for polynomial ideals, connecting them to Lasserre's relaxations and providing new tools for combinatorial optimization.

Findings

Theta bodies form a nested hierarchy of semidefinite relaxations.

The first theta body coincides with Lovász's theta body for graphs.

Explicit computation of theta bodies using combinatorial moment matrices.

Abstract

Inspired by a question of Lov\'asz, we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal, called theta bodies of the ideal. For the stable set problem in a graph, the first theta body in this hierarchy is exactly Lov\'asz's theta body of the graph. We prove that theta bodies are, up to closure, a version of Lasserre's relaxations for real solutions to ideals, and that they can be computed explicitly using combinatorial moment matrices. Theta bodies provide a new canonical set of semidefinite relaxations for the max cut problem. For vanishing ideals of finite point sets, we give several equivalent characterizations of when the first theta body equals the convex hull of the points. We also determine the structure of the first theta body for all ideals.