SOS lower bounds with hard constraints: think global, act local

TL;DR

This paper improves SOS lower bounds for CSPs by translating global constraints into local ones, demonstrating limitations of high-degree SOS in approximating Min-Bisection and Max-Bisection problems.

Contribution

It introduces a method to handle global constraints within SOS lower bounds, enabling new bounds for problems like Min-Bisection and Max-Bisection.

Findings

Degree-Ω(√n) SOS cannot approximate Min-Bisection within 4/3 - ε.

Degree-Ω(n) SOS cannot approximate Max-Bisection within 11/12 + ε.

New SOS lower bounds for problems previously without such results.

Abstract

Many previous Sum-of-Squares (SOS) lower bounds for CSPs had two deficiencies related to global constraints. First, they were not able to support a "cardinality constraint", as in, say, the Min-Bisection problem. Second, while the pseudoexpectation of the objective function was shown to have some value $\beta$, it did not necessarily actually "satisfy" the constraint "objective = $\beta$". In this paper we show how to remedy both deficiencies in the case of random CSPs, by translating \emph{global} constraints into \emph{local} constraints. Using these ideas, we also show that degree-$\Omega(\sqrt{n})$ SOS does not provide a $(\frac{4}{3} - \epsilon)$-approximation for Min-Bisection, and degree-$\Omega(n)$ SOS does not provide a $(\frac{11}{12} + \epsilon)$-approximation for Max-Bisection or a $(\frac{5}{4} - \epsilon)$-approximation for Min-Bisection. No prior SOS lower bounds for these problems were known.