Optimization over the Boolean Hypercube via Sums of Nonnegative Circuit Polynomials

TL;DR

This paper explores the use of sums of nonnegative circuit polynomials (SONC) as an alternative to sums of squares (SOS) for polynomial optimization over the boolean hypercube, establishing key theoretical parallels.

Contribution

It introduces SONC certificates for boolean hypercube optimization, showing they can match SOS results in degree bounds and certificate complexity.

Findings

SONC certificates exist with degree at most n+d for nonnegative polynomials.

Short SONC certificates with at most n^{O(d)} terms are possible.

Key SOS-based complexity bounds also hold for SONC certificates.

Abstract

Various key problems from theoretical computer science can be expressed as polynomial optimization problems over the boolean hypercube. One particularly successful way to prove complexity bounds for these types of problems are based on sums of squares (SOS) as nonnegativity certificates. In this article, we initiate the analysis of optimization problems over the boolean hypercube via a recent, alternative certificate called sums of nonnegative circuit polynomials (SONC). We show that key results for SOS based certificates remain valid: First, for polynomials, which are nonnegative over the $n$-variate boolean hypercube with constraints of degree $d$ there exists a SONC certificate of degree at most $n+d$. Second, if there exists a degree $d$ SONC certificate for nonnegativity of a polynomial over the boolean hypercube, then there also exists a short degree $d$ SONC certificate, that includes at most $n^{O(d)}$ nonnegative circuit polynomials.