Sum-of-squares hierarchy lower bounds for symmetric formulations

TL;DR

This paper develops a symmetry-based method to establish lower bounds for the Sum-of-Squares hierarchy, simplifying the analysis to univariate polynomial inequalities, and applies it to problems like the integer cut polytope and Min-Knapsack.

Contribution

It introduces a novel technique leveraging symmetry to prove SoS hierarchy lower bounds, simplifying the analysis to univariate polynomial inequalities and demonstrating its effectiveness on specific problems.

Findings

Proves a lower bound for the integer cut polytope using a simplified approach.

Shows the SoS hierarchy needs many rounds to improve the Min-Knapsack gap.

Provides elementary proofs for known lower bounds using symmetry-based methods.

Abstract

We introduce a method for proving Sum-of-Squares (SoS)/ Lasserre hierarchy lower bounds when the initial problem formulation exhibits a high degree of symmetry. Our main technical theorem allows us to reduce the study of the positive semidefiniteness to the analysis of "well-behaved" univariate polynomial inequalities. We illustrate the technique on two problems, one unconstrained and the other with constraints. More precisely, we give a short elementary proof of Grigoriev/Laurent lower bound for finding the integer cut polytope of the complete graph. We also show that the SoS hierarchy requires a non-constant number of rounds to improve the initial integrality gap of 2 for the Min-Knapsack linear program strengthened with cover inequalities.