SoS certification for symmetric quadratic functions and its connection to constrained Boolean hypercube optimization

TL;DR

This paper investigates the Sum of Squares hierarchy's rank for symmetric quadratic functions on the Boolean hypercube, refutes a prior conjecture of linear lower bounds, and provides new upper bounds connecting to combinatorial optimization problems.

Contribution

It refutes a conjecture that the SoS rank is linear in n for symmetric quadratic functions, establishing an upper bound of O(√(nk) log n) using Chebyshev polynomials.

Findings

Refutes the conjecture of linear SoS rank lower bound for SQFs.

Establishes an upper bound of O(√(nk) log n) for the SoS rank of SQFs.

Provides new SoS certificates for Min Knapsack and Set Cover problems.

Abstract

We study the rank of the Sum of Squares (SoS) hierarchy over the Boolean hypercube for Symmetric Quadratic Functions (SQFs) in $n$ variables with roots placed in points $k-1$ and $k$. Functions of this type have played a central role in deepening the understanding of the performance of the SoS method for various unconstrained Boolean hypercube optimization problems, including the Max Cut problem. Recently, Lee, Prakash, de Wolf, and Yuen proved a lower bound on the SoS rank for SQFs of $\Omega(\sqrt{k(n-k)})$ and conjectured the lower bound of $\Omega(n)$ by similarity to a polynomial representation of the $n$-bit OR function. Using Chebyshev polynomials, we refute the Lee -- Prakash -- de~Wolf -- Yuen conjecture and prove that the SoS rank for SQFs is at most $O(\sqrt{nk}\log(n))$. We connect this result to two constrained Boolean hypercube optimization problems. First, we provide a degree $O( \sqrt{n})$ SoS certificate that matches the known SoS rank lower bound for an instance of Min Knapsack, a problem that was intensively studied in the literature. Second, we study an instance of the Set Cover problem for which Bienstock and Zuckerberg conjectured an SoS rank lower bound of $n/4$. We refute the Bienstock -- Zuckerberg conjecture and provide a degree $O(\sqrt{n}\log(n))$ SoS certificate for this problem.