A new semidefinite programming hierarchy for cycles in binary matroids and cuts in graphs

TL;DR

This paper introduces a new semidefinite programming hierarchy based on theta bodies for cycles in binary matroids and applies it to cuts in graphs, providing new relaxations and characterizations of the cut polytope.

Contribution

It constructs theta bodies for the vanishing ideal of cycles in binary matroids and develops a hierarchy of relaxations for the cut polytope in graphs.

Findings

First theta body equals the cycle polytope under certain minor exclusions.

New hierarchy improves relaxation of the cut polytope.

Solves a problem posed by Lovász regarding cycle polytopes.

Abstract

The theta bodies of a polynomial ideal are a series of semidefinite programming relaxations of the convex hull of the real variety of the ideal. In this paper we construct the theta bodies of the vanishing ideal of cycles in a binary matroid. Applied to cuts in graphs, this yields a new hierarchy of semidefinite programming relaxations of the cut polytope of the graph. If the binary matroid avoids certain minors we can characterize when the first theta body in the hierarchy equals the cycle polytope of the matroid. Specialized to cuts in graphs, this result solves a problem posed by Lov\'asz.