Tight Sum-of-Squares lower bounds for binary polynomial optimization problems

TL;DR

This paper establishes tight lower bounds on the levels of the Sum-of-Squares hierarchy needed for binary polynomial optimization and disproves a conjecture regarding the hierarchy's rank for certain integer hull problems.

Contribution

It provides exact lower bounds for the SoS hierarchy's levels in binary polynomial optimization and refutes a conjecture about the hierarchy's rank for detecting integer hulls.

Findings

$rac{n+2d-1}{2}$ levels are necessary for exact solutions in certain problems

Disproves Laurent's conjecture that SoS rank is $n-1$ for detecting empty integer hulls

Establishes bounds matching recent upper bounds for the hierarchy's power

Abstract

We give two results concerning the power of the Sum-of-Squares(SoS)/Lasserre hierarchy. For binary polynomial optimization problems of degree $2d$ and an odd number of variables $n$, we prove that $\frac{n+2d-1}{2}$ levels of the SoS/Lasserre hierarchy are necessary to provide the exact optimal value. This matches the recent upper bound result by Sakaue, Takeda, Kim and Ito. Additionally, we study a conjecture by Laurent, who considered the linear representation of a set with no integral points. She showed that the Sherali-Adams hierarchy requires $n$ levels to detect the empty integer hull, and conjectured that the SoS/Lasserre rank for the same problem is $n-1$. We disprove this conjecture and derive lower and upper bounds for the rank.