Definable Ellipsoid Method, Sums-of-Squares Proofs, and the Graph Isomorphism Problem

TL;DR

This paper demonstrates that the feasibility problem for semidefinite programs can be expressed in fixed-point logic with counting, and shows that the Lasserre/Sums-of-Squares hierarchy collapses to Sherali-Adams for graph isomorphism.

Contribution

It extends the logical definability of the ellipsoid method to bounded convex sets and connects this to the collapse of SDP hierarchies in graph isomorphism.

Findings

Feasibility of semidefinite programs is expressible in infinitary FPC.

Lasserre/Sums-of-Squares hierarchy collapses to Sherali-Adams hierarchy.

Ellipsoid method applicability extends to explicitly bounded convex sets.

Abstract

The ellipsoid method is an algorithm that solves the (weak) feasibility and linear optimization problems for convex sets by making oracle calls to their (weak) separation problem. We observe that the previously known method for showing that this reduction can be done in fixed-point logic with counting (FPC) for linear and semidefinite programs applies to any family of explicitly bounded convex sets. We use this observation to show that the exact feasibility problem for semidefinite programs is expressible in the infinitary version of FPC. As a corollary we get that, for the isomorphism problem, the Lasserre/Sums-of-Squares semidefinite programming hierarchy of relaxations collapses to the Sherali-Adams linear programming hierarchy, up to a small loss in the degree.